Chapter 8
A Model for Microstructure
Characterization
8.1 Introduction
Titanium is noisy. Contrary to the situation discussed in Chap. 7 in which the
surface of an isotropic body, copper, is made artificially noisy through shotpeening, titanium is intrinsically noisy because of its microstructure consisting of
a random distribution of anisotropic crystallites. Figure 8.1 illustrates this condition
by comparing the response of the same flaw in the alloy, Ti-6Al-4V, to aluminum,
which is isotropic.
Figure 8.2 contrasts noise in a titanium alloy with a large crack response. The
origin of the noise lies in the random, anisotropic microstructure of the alloy, and
the objective of the present chapter is to develop a model for computing this noise,
starting with a realistic model of the microstructure.
Figures 8.3 and 8.4 further illustrate the microstructure of a titanium alloy, in
this case, Ti-7Al. The clustering of grains of similar orientation in Fig. 8.4 produces
continuum effects that are different than those of the individual grains. We expect
to see conductivity variations that are smoothed and elongated in the direction of
the clusters, rather than chaotic from grain to grain. Thus, the grain size used for a
particular property model is no longer actually the grain size measured from optical
or backscattered electron (BSE) images.
8.2 Stochastic Euler Space
In Appendix 1 of Chap. 7 we pointed out that the anisotropic model was an
approximate one. In this chapter we intend to rectify that, and we will extend the
work of Chap. 7 to two dimensions, with the intention of developing an approach to
characterizing the microstructure of Ti and its alloys. We’ll start with Euler Angles.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
H. A. Sabbagh et al., Advanced Electromagnetic Models for Materials
Characterization and Nondestructive Evaluation, Scientific Computation,
https://doi.org/10.1007/978-3-030-67956-9_8
197
A Model for Microstructure
Characterization
8.1 Introduction
Titanium is noisy. Contrary to the situation discussed in Chap. 7 in which the
surface of an isotropic body, copper, is made artificially noisy through shotpeening, titanium is intrinsically noisy because of its microstructure consisting of
a random distribution of anisotropic crystallites. Figure 8.1 illustrates this condition
by comparing the response of the same flaw in the alloy, Ti-6Al-4V, to aluminum,
which is isotropic.
Figure 8.2 contrasts noise in a titanium alloy with a large crack response. The
origin of the noise lies in the random, anisotropic microstructure of the alloy, and
the objective of the present chapter is to develop a model for computing this noise,
starting with a realistic model of the microstructure.
Figures 8.3 and 8.4 further illustrate the microstructure of a titanium alloy, in
this case, Ti-7Al. The clustering of grains of similar orientation in Fig. 8.4 produces
continuum effects that are different than those of the individual grains. We expect
to see conductivity variations that are smoothed and elongated in the direction of
the clusters, rather than chaotic from grain to grain. Thus, the grain size used for a
particular property model is no longer actually the grain size measured from optical
or backscattered electron (BSE) images.
8.2 Stochastic Euler Space
In Appendix 1 of Chap. 7 we pointed out that the anisotropic model was an
approximate one. In this chapter we intend to rectify that, and we will extend the
work of Chap. 7 to two dimensions, with the intention of developing an approach to
characterizing the microstructure of Ti and its alloys. We’ll start with Euler Angles.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
H. A. Sabbagh et al., Advanced Electromagnetic Models for Materials
Characterization and Nondestructive Evaluation, Scientific Computation,
https://doi.org/10.1007/978-3-030-67956-9_8
197