Determination of texture coefficients in hexagonal polycrystalline aggregates with orthorhombic symmetry using ultrasounds

Abstract

A method to characterize the texture of hexagonal polycrystalline aggregates with orthorhombic macroscopic symmetry is presented. Previous methods are extended to the analysis of waves propagating on three principal planes of the material. Five coefficients of the crystallographic orientation distribution function, W20o, W2:0, W•0o, W4:0, and W440, are determined from angular variations of the phase velocity of the five ultrasonic modes considered: Bulk longitudinal waves, shear waves with in-plane and plane-normal polarizations, Rayleigh surface waves, and the fundamental symmetric plate mode (in the long-wavelength limit). Analytical expressions for the texture coefficients are found for an arbitrary number of principal planes. Measurements are reported for a Zr-2.5% wt Nb pressure tube, an alloy used in CANDU nuclear reactors. The angular variations of the phase velocity of leaky Rayleigh waves and leaky surface skimming compressional waves are measured with a line-focus acoustic microscope. Although these waves are not exactly those treated in the theory, they are closely related to pure Rayleigh waves and bulk longitudinal waves. The determined texture coefficients are sensitive to the input physical parameters when certain combinations of planes and acoustic waves requiring absolute velocity measurements are used. However, the coefficients are less sensitive to the input parameters when combinations requiring only relative velocity measurements are used. Results derived from relative measurements agreed with neutron scattering measurements.

A spherical harmonic approach for the determination of HCP texture from ultrasound: A solution to the inverse problem

“…Second and more importantly, when performing inverse studies (see later), since the best accuracy ultrasonic wave velocity tests can possibly achieve is normally worse than ±3.79 m/s (typically ±10 m/s), it becomes apparent that it is not possible to recover 6th or higher degree texture SH coefficients from ultrasonic wave velocity tests, because under these circumstances, the introduced numerical noise is already much larger than the SH coefficient information to be extracted. This explains mathematically why it has been widely reported in the literature that only texture coefficients up to 4th degree are obtained [26,31,32,34,35]. Note that the SH coefficients arising in Tables 1 -4 relate to HCP Ti-6Al-4V alpha phase, but that in general, the order of SH coefficients theoretically obtainable is determined by the spherical expansion of a given material's single crystal elastic properties.…”

“…These approaches laid down the theoretical foundations for the majority of the subsequent studies, especially for texture detection on plate-shaped samples where the texture relations with other wave modes like Rayleigh and Lamb waves can also be utilized [30][31][32]. For example, Sayers [33] derived the expressions for angular variation of ultrasonic waves in hexagonal materials for non-destructive tests;…”

“…Hairo [31]combined the texture relations with both Rayleigh surface waves and skimming Lamb wave modes to get the non-zero ODF coefficients and obtained coarse pole figures; Kielczynski et al [32] made use of bulk longitudinal wave, Rayleigh surface wave and Lamb wave modes to obtain five texture coefficients for hexagonal samples with orthorhombic symmetry; Thompson et al [30] and Dixon et al [34] used only the Lamb wave dependence of texture coefficients to obtain three 4th-order ODF coefficients for cubic materials. Anderson et al [35] performed similar studies to extract Kearns factors from hexagonal-grained Zircaloys.…”

Correlation between elastic anisotropy and magnetic susceptibility anisotropy of sedimentary and metamorphic rock

“…It is interesting that, historically, the importance of the ordering of crystallite positions was first revealed in rock; only then, the notion of "texture" was introduced (see [7]). There are a large number of works on acoustics in which the appearance of elastic anisotropy has been proved after mechanical processing of initially isotro pic polycrystalline materials with chaotic grain orien tation (see, e.g., [8,9]). It is noted in [7] that elastic anisotropy is formed due to inhomogeneous plastic strains inside a polycrystalline material, which yields variations in the shape of individual crystallites and/or their rotation, i.e., the appearance of macroscopic ordering and, as the result, anisotropy of acoustic characteristics.…”

Ultrasonic texture characterization of aluminum, zirconium and titanium alloys

“…The final solutions for waves in hexagonal polycrystals are presented in equation 14^® and a schematic of the displacements occurring in this wave type is shown in Figure 2. Likewise, SHo wave solutions for hexagonal polycrystals are presented in equation (15)^® with a schematic of the displacements shown in Figure 2. Also shown in Figure 2 is a schematic of the measurement procedure for both of these wave types.…”