Ultrasonic attenuation and backscattering in polycrystalline materials with nonspherical grains

An ultrasonic wave propagating through a microscopically inhomogeneous medium, such as polycrystalline materials, is subject to scattering at the grain boundaries as well as other inhomogenities. The fraction of energy removed from the incident wave is responsible for important phenomenon like attenuation, dispersion, and background “noise" associated with a given ultrasonic inspection system. Quantitative knowledge of attenuation, phase velocity, and scattered wave field are extremely important for a reliable nondestructive evaluation of such materials. Expected propagation characteristics of ultrasonic waves in randomly oriented equiaxed grains are fairly well understood. But when the grains are elongated and/or preferentially oriented, the wave propagation constants exhibit anisotropic behavior. The present paper sheds more light on the effect of grain shape on the attenuation and dispersion of ultrasonic waves in polycrystals. Specifically, theoretical results are presented showing the effects of different grain aspect ratios. It is observed that for the same effective grain volume, grain elongation has smaller effect on attenuation. Although considerable attention has been given to the understanding of mean propagation characteristics of an ultrasonic beam, until recently, there have been relatively little efforts devoted towards rigorous treatments of backscattered signals from the material microstructure. In this paper, we also attempt to include some degree of multiple scattering in the calculation of the backscattered signals by developing a formalism that relates mean wave propagation characteristics to the noise. INTRODUCTION A polycrystalline material is composed of numerous discrete grains, each having a regular, crystalline atomic structure. The elastic properties of the grains are anisotropic and their crystallographic axes are oriented differently. When an acoustic wave propagates through such a polycrystalline aggregate, it is attenuated by scattering at the grain boundaries, with the value of this attenuation and the related shift in the propagation velocity depending on the size, shape, orientation distributions, and crystalline anisotropy of the grains. If the grains are equiaxed and randomly oriented, these propagation properties are independent of direction, but such is not the case when the grains are elongated and/or have preferred crystallographic orientation. Therefore, reliable ultrasonic testing of engineering alloy components require the knowledge of the anisotropies in the attenuation and velocities of ultrasonic waves due to preferred grain orientations and elongated shapes. The propagation of elastic waves in randomly oriented, equiaxed polycrystals has received considerable attention, with most recent contributions for the cubic materials being made by Hirsekorn [1,2] Stanke and Kino [3,4], Beltzer and Brauner [5], and Turner [6]. Stanke and Kino present their ``unified theory" based on the second order Keller approximation [7] and the use of a geometric autocorrelation function to describe the grain size distribution. Stanke and Kino argue that their approach is to be preferred because i) the unified theory more fully treats multiple scattering, ii) the unified theory avoids the high frequency oscillations which are coherent artifacts of the single-sized, spherical grains assumed by Hirsekorn, and iii) the unified theory correctly captures the high frequency “geometric regime” in which the Born approximation breaks down. The theoretical treatment of ultrasonic wave propagation in preferentially oriented grains is more limited. Hirsekorn has extended her theory to the case of preferred crystallographic orientation while retaining the assumption of spherical grain shape [8], and has performed numerical calculations for the case of stainless steel with fully aligned [001] axes [9]. Turner, on the other hand, derives the Dyson equation using anisotropic Green's functions to predict the mean ultrasonic field in macroscopically anisotropic medium [6]. He then proceeds to obtain the solution of the Dyson equation for the case of equiaxed grains with aligned [001] axes. Previously we have employed the formalism of Stanke and Kino [3,4] in [001] aligned stainless steel polycrystal to compute the mean attenuation and phase velocity of plane ultrasonic waves [10,11]. In this paper we revisit our earlier calculations for the case of elongated grains and focus our attention on the effect of grain shape on the mean propagation characteristics. Specifically, we consider two cases: 1) the [001] crystallographic axes are aligned with the z-axis of the laboratory coordinate system while remaining two axes are randomly oriented and 2) all the crystallographic axes are randomly oriented. In both cases, the crystallites have cubic symmetry and the grains are considered to be ellipsoidal with either their major or minor axis parallel to the z-direction of the laboratory coordinate system. Numerical results for the attenuation and phase velocity of longitudinal wave in these two polycrystals are presented here. The material properties of the two media are listed in Table 1. Until recently, there have been fewer attempts to develop rigorous expressions for backscattered signals. Margetan et. al. [21] formulated backscattered power using independent scatterer approximation. More recently, Rose [22] has developed a general formalism, based on Auld's [23] electro-mechanical reciprocity relations. Since this formalism is basically intractable, he then proceeded with the relevant calculations using Born and the single scattering approximations. We have, in the past [24], employed Rose's formalism to calculate the backscattered power due to preferentially oriented spherical and nonspherical grains. In this paper, we describe a formalism that accounts for some degree of multiple scattering in the calculation of the backscattered signals. Computed results for the cases of randomly oriented equiaxed and elongated grains are also presented in this paper. Table 1. Material Properties Material 11 c ( 2 / m N ) 12 c ( 2 / m N ) 44 c ( 2 / m N ) ρ ( 3 / m kg ) Iron 21.6 10 10 × 14.5 10 10 × 12.9 10 10 × 7.86 3 10 × THEORY Mean wave propagation The displacement field due to an ultrasonic wave propagating in a polycrystalline material can be described by the stochastic wave equation 0 ) ( ) ( )] ( ) ( [ 2 , = + r u r r u r C i j kl ijkl ξ ξ ξ ξ ω ρ , (1) where ) (r C ijkl ξ is the actual local elastic tensor, ) (r ξ ρ is the actual local density, and ) (r u i ξ is the actual displacement field in the medium ξ. The set of elastic tensors and the probability density function ) (ξ p , which is the probability of choosing any particular medium, form a stochastic process. In a medium with no density variation, the application of the unified theory of Stanke and Kino [3] to the wave equation yields the generalized following Christoffel's equation for the expected propagation constant k. 0 ] / [ 2 2 = − Γ ik ik k δ ρω (2a)