Ultrasonic Propagation and Scattering in Duplex Microstructures with Application to Titanium Alloys

Abstract

A general model for elongated duplex microstructures is proposed for modeling scattering-induced ultrasonic longitudinal attenuation in hexagonal polycrystals for application to titanium alloys. The material system consists of microtextured regions (MTRs) which are formed by much smaller α crystallites with preferred orientations. Their preferred orientation is represented by a modified Gaussian orientation distribution function. Scattering induced by MTRs and by crystallites is added to obtain ultrasonic attenuation in the medium. The effective elastic properties of MTRs are determined and used to obtain the scattering-induced MTR attenuation and backscattering. Crystallite attenuation is estimated by the untextured attenuation coefficient factored by a texture transition function. The total attenuation is obtained by combining solutions for microtextured region attenuation and crystallite attenuation. Spectroscopic attenuation and backscattering measurements are performed on a forged sample of titanium alloy. Reasonable agreement is found between experiment and the model predictions with a given texture parameter.

Appendix: Attenuation Coefficients for Hexagonal Elongated Grains

In this appendix we briefly describe the computation algorithm for the attenuation coefficients of the medium with hexagonal elongated grains. Given the geometrical two point correlation function of ellipsoidal [5, 14, 17–19] grains as a generalized exponential function in terms of three effective ellipsoid radii in the form

$$ w( x,y,z ) = \exp\biggl( - \sqrt{\frac{x^{2}}{a_{x}^{2}} + \frac{y^{2}}{a_{y}^{2}} + \frac{z^{2}}{a_{z}^{2}}} \biggr), $$

(A.1)

where a _x ,a _y and a _z are the effective ellipsoid radii for three major axes corresponding to x, y and z coordinates (Fig. 1(b)), the nondimensional attenuation coefficients for hexagonal and elongated crystallites are expressed as

(A.2)

where α _QM are attenuation coefficient components due to scattering of wave mode Q to wave mode M. As described in detail in [30] and similar to the approach developed for cubic elongated crystallites [5], we obtain for each attenuation component α _QM

(A.3)

where $S_{QM}^{0}$, $S_{QM}^{1}$ and $S_{QM}^{2}$ were given as

(A.4)

and

V _L , V _T are macroscopic velocities obtained by Voigt averaging [10] (for MTRs they are obtained from the effective elastic constants); it is important to note that as a result of averaging those velocities are independent of the parameter σ and are thus equal to those used for the second phase (crystallites).

Using the analysis of [13] for the hexagonal crystallites the inner products are written in the form

(A.5)

where the coefficients A _QM , B _QM , and C _QM were expressed in terms of A, B and D in the following:

(A.6)

and

$$ X = \hat{\mathbf{p}} \cdot\hat{\mathbf{s}} = \cos(\phi - \phi_{\tau} )\sin\theta\sin\tau + \cos\theta\cos\tau. $$

(A.7)

The wave propagation and scattering directions are defined in the coordinate system as shown in Fig. 1(b) by

For the attenuation term α _MTR in Eq. (11) the coefficients A=A _MTR , B=B _MTR , D=D _MTR in Eq. (A.6) are obtained from Eq. (10).