2.3.2. Stress Determination

Based on the entire diffraction spectrum, residual stresses can be calculated either from the change in lattice parameters of each phase or from individual *hkl* peak shifts. The former represents the average stress behavior of each phase (called mean phase stress), and the latter represents the stress in individual grains (called peak-specific residual stress).

Mean Phase Stresses

The mean phase strain, given by the weighted average of several single-peak strains, was determined from the change in the average lattice parameters of each phase in composites with respect to those in stress-free reference powders [28]. In this case, all diffraction peaks were considered, thus allowing the representation of residual stresses more precisely and reliably for bulk composites.

In this study, the strain for the hexagonal *α*-Al2O<sup>3</sup> phase can be calculated along lattice axes *a* and *c*:

$$
\varepsilon\_d = \frac{a - a\_0}{a\_0} \tag{1}
$$

$$
\varepsilon\_{\mathcal{C}} = \frac{c - c\_0}{c\_0} \tag{2}
$$

where *a* (*a* = *b*) and *c* are the lattice parameters of the *α*-Al2O<sup>3</sup> phase, and *a*<sup>0</sup> and *c*<sup>0</sup> are the stress-free lattice parameters measured from the Al2O<sup>3</sup> starting powder. For orthorhombic AT, the strains in the *a*, *b*, and *c* axes can be similarly determined.

Without considering granular anisotropy, the mean phase strains of the Al2O<sup>3</sup> and AT phases in gauge volume were calculated by averaging the strain over the unit cell, as presented in [32].

$$
\overline{\varepsilon}\_A = \frac{1}{3} (2\varepsilon\_a + \varepsilon\_c)\_A \tag{3}
$$

$$\mathbb{E}\_{AT} = \frac{1}{3} (\varepsilon\_a + \varepsilon\_b + \varepsilon\_c)\_{AT} \tag{4}$$

Considering the fabrication process and geometrical symmetry of samples, the strains were assumed to be isotropic in a direction parallel to the major plane of samples. Thus, the strains measured in the in-plane and normal directions are sufficient for stress determination. These are used as principal strains *<sup>ε</sup>ii* (*<sup>i</sup>* = 1, 2, 3), *<sup>ε</sup>*<sup>11</sup> = *<sup>ε</sup>*<sup>22</sup> = *<sup>ε</sup>In*−*plane*, and *ε*<sup>33</sup> = *εNormal*. Thus, the mean phase stresses of Al2O<sup>3</sup> and AT were determined in the in-plane and normal directions using Hooke's law, respectively, as follows:

$$
\sigma\_{In-plane} = \frac{E}{1+\nu} \overline{\varepsilon}\_{In-plane} + \frac{E\nu}{(1+\nu)(1-2\nu)} \left(2\overline{\varepsilon}\_{In-plane} + \overline{\varepsilon}\_{Normal}\right) \tag{5}
$$

$$
\sigma\_{\text{Normal}} = \frac{E}{1+\nu} \overline{\varepsilon}\_{\text{Normal}} + \frac{E\nu}{(1+\nu)(1-2\nu)} \left(2\overline{\varepsilon}\_{\text{In-plane}} + \overline{\varepsilon}\_{\text{Normal}}\right) \tag{6}
$$

where *ε* corresponds to the calculated mean phase strains of Al2O<sup>3</sup> and AT phases, as given by Equations (3) and (4), respectively; and *E* and *ν* are the bulk elastic constants of Al2O<sup>3</sup> (*E* = 400 GPa and *ν* = 0.22) [33] and AT (*E* = 284 GPa and *ν* = 0.33) [9].
