Inversion of Ultrasonic Attenuation for Textural Information of Polycrystals

The mechanical properties of polycrystalline materials depend on the individual physical properties of the constituent grains. When grains are randomly oriented with respect to a fixed coordinate system, the average elastic properties are isotropic. A polycrystalline aggregate possesses macroscopic texture, and thus exhibit anisotropic elastic properties, when the grains are preferentially oriented. Accurate knowledge of texture is important for a number of engineering applications. For example, it plays an important role in determining their subsequent formability into finished parts of complex shape by deep drawing. Textural information can also be exploited to assess plastic damage in a component due to fatigue or external impact. Current ultrasonic methods are based on the relationship between ultrasonic phase velocities and only the low-order orientation distribution coefficients (ODCs, a set of orthonormal basis functions to express crystallographic orientation of a grain), a consequence of the application of Voigt averaging method to infer aggregate elastic properties. Thus, such techniques provide only a partial description of texture. It is well-known from the scattering theories that while phase velocity of acoustic waves is controlled primarily by averages of single grain elastic constant fluctuations, attenuation due to scattering of such waves at the grain boundaries are controlled by two-point averages of these fluctuations. Since these two-point averages depend on higher order ODCs, inversion of measured attenuation is expected to yield a more complete and accurate description of texture. In this paper, we will present computed frequency and direction dependent attenuation coefficient of ultrasonic waves in orthotropic polycrystals with equiaxed cubic grains and mathematically investigate the inversion of computed attenuation coefficient to obtain higher order ODCs. We will also compute pole figures and compare them with those obtained by current ultrasonic methods. 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 Ahmed and Thompson [10, 11] 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. The effects of grain scattering on mean/expected acoustic waves are controlled by the so-called oneand two-point averages which depend on the orientation distribution of the individual grains. It is customary to represent these orientations by the ODCs [12]. In this paper we first compute the expected wave vector in a macroscopically orthotropic polycrystalline material with cubic grains. Next, we proceed to show how one can proceed to invert the attenuation data for twelve ODCs by using the computed attenuation. We choose polycrystalline iron for our calculations with orthotropic symmetry. Table 1 shows the single crystal physical properties. The twelve ODCs employed in our calculations are listed in Table 2. 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 Table 2. Orientation distribution coefficients l = 4 l = 6 l = 8 00 l W 3 10 702024 . 6 − × − 3 10 7245817 . 4 − × 4 1