# Application of Laser-Ultrasonics for Evaluating Textures and Anisotropy
## 1 Introduction
There are indications that dislocations, existing after cold working, may somewhat reduce the elastic stiffness [5] and the elongated grains may produce an anisotropy additional to that from texture, but these contributions seem to be fairly small [8; 20]. In most earlier work, the full quantification of texture was carried out using polycrystalline diffraction, i.e., pole figure methods, but these are not sufficient for evaluating the elastic tensors or matrices of the materials. It is necessary, first, to calculate orientation distribution functions (ODFs), which is often achieved using spherical harmonic functions, e.g., [21] although other mathematical procedures are also possible and have been applied in some cases, e.g., [1]. An advantage of the spherical harmonic approach is that the linear elastic behaviour can be completely described using only three coefficients. These are the C\({}_{4}\)11, C\({}_{4}\)12, and C\({}_{4}\)13 in the Bunge notation [21] or the equivalent W\({}_{400}\), W\({}_{420}\), W\({}_{440}\) coefficients in the Roe notation [22]. More recently, electron back-scattering diffraction (EBSD) has become widely adopted for texture measurements, which permits the complete orientations of individual crystals to be measured and not only polycrystalline pole figures. Data from EBSD can be fitted to the same ODF functions, but this is not essential for calculating physical properties, which can be achieved directly from the individual crystal data. Both methods lead to the same results. When considering the inverse procedure of determining texture from ultrasonic velocities and elasticity matrices, it is evident that the measurements must elucidate the anisotropic character of the material, so a single measurement cannot suffice. The necessary conditions can, however, be achieved in different ways in practice. 1. Perhaps the most obvious approach is to measure the same property in different directions by rotating the material, with respect to the direction of measurement in the instrument, as described in several publications, e.g., [1; 4; 5; 6; 8; 10; 11; 13; 16; 23]. We also demonstrate a method further below by which this can be achieved with high precision. 2. Another approach is to determine the velocities for different wave types propagating along the same direction. This has been demonstrated by using combinations of S\({}_{0}\) and S\({}_{H0}\) waves [5] or S\({}_{0}\), S\({}_{H0}\) and P-waves [7; 9] and has been applied in steelworks for measuring plastic anisotropy ratios (r-values), using EMATs, under production conditions for (cold) annealed sheets. 3. A variant based on the previous concept using resonances of P- and S- waves through the thickness of plates has also been tested under steelwork conditions [12]. 4. Another approach used P-waves arrivals measured after different numbers of reflections through the thickness of the plate [15]. Thus, although the same fixed positions were used for generation and detection, the waves passed along different directions in the material. 5. A LUS variant [14] based on the first method, here, was to convert the generating laser spot into a circle and mask off different sectors so that measurements of P- and SAW-waves could be made in different directions, which was reported to give robust measurements. 6. Additionally, the group at Nottingham recently developed the Spatially Resolved Acoustic Spectroscopy (SRAS) technique [24] with which the surface texture can be measured by sampling the orientation of individual grains. From the measured surface, the texture can be calculated in a similar manner to how the textures from EBSD measurements are calculated. The team has recently been able to perform the inverse calculation of the single crystal values from the SRAS measurements [25]. In the present paper, we demonstrate a refinement of the first method and also an approach that resembles the fifth method, but where the generating laser beam is deflected using Galvano mirrors.