Peak-Specific Residual Stresses

Owing to single-crystal anisotropy, the strain measured by diffraction varies among grain families; consequently, such is the case among diffraction peaks [34]. The deviation from the average phase stress in a given grain, that is, the peak-specific residual stresses, can be obtained by measuring the shifts in individual diffraction peaks. Understanding peak-specific residual stresses is important because the stress concentrations at the scale of individual grains may ultimately affect crack initiation processes in fatigue or brittle failure.

In TOF diffraction, at a fixed scattering angle, 2*θB*, the lattice spacing, *dhkl*, of the *hkl* plane can be obtained from the TOF of the peak position, *thkl* [23]:

$$d\_{hkl} = \frac{h}{2\sin\theta\_B mL} t\_{hkl} \tag{7}$$

where *h* is the Planck constant, *m* is the neutron mass, and *L* is the neutron flight path length.

The peak-specific residual strain, which depends on the change in the lattice spacing of the ∆*dhkl hkl* plane, can then be calculated in terms of the TOF shift in the recorded peak, ∆*thkl*:

$$
\varepsilon\_{\text{hkl}} = \Delta d\_{\text{hkl}} / d\_{\text{hkl}}^0 = \Delta t\_{\text{hkl}} / t\_{\text{hkl}}^0 \tag{8}
$$

where *d* 0 *hkl* is the strain-free reference lattice spacing, *t* 0 *hkl*, and strain-free reference TOF at the *hkl* peak. The peak positions can be precisely determined by single-peak fitting, with a typical sensitivity of *<sup>δ</sup><sup>ε</sup>* <sup>=</sup> <sup>∆</sup>*dhkl*/*dhkl* <sup>∼</sup><sup>=</sup> <sup>50</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> . The peak-specific residual strains measured along the in-plane and normal directions were characterized as *ε hkl In*−*plane* and *ε hkl Normal*, respectively. They were considered as principal strain components *εii* (*i* = 1, 2, 3; *<sup>ε</sup>*<sup>11</sup> = *<sup>ε</sup>*<sup>22</sup> = *<sup>ε</sup>In*−*plane*; and *<sup>ε</sup>*<sup>33</sup> = *<sup>ε</sup>Normal*).

The peak-specific residual stresses for each reflection of Al2O<sup>3</sup> and AT in the in-plane and normal directions (*σ hkl In*−*plane* and *<sup>σ</sup> hkl Normal*) were calculated using Hooke's law, as follows:

$$\sigma\_{\text{In}-plane}^{\text{hkl}} = \frac{E\_{\text{hkl}}}{1 + \nu\_{\text{hkl}}} \varepsilon\_{\text{In}-plane}^{\text{hkl}} + \frac{E\_{\text{hkl}} \upsilon\_{\text{hkl}}}{(1 + \nu\_{\text{hkl}})(1 - 2\nu\_{\text{hkl}})} \left(2\varepsilon\_{\text{In}-plane}^{\text{hkl}} + \varepsilon\_{\text{Normal}}^{\text{hkl}}\right) \tag{9}$$

$$\sigma\_{\text{Normal}}^{\text{hkl}} = \frac{\mathbb{E}\_{\text{hkl}}}{1 + \nu\_{\text{hkl}}} \varepsilon\_{\text{Normal}}^{\text{hkl}} + \frac{\mathbb{E}\_{\text{hkl}} \nu\_{\text{hkl}}}{(1 + \nu\_{\text{hkl}})(1 - 2\nu\_{\text{hkl}})} \left(2\varepsilon\_{\text{In-plane}}^{\text{hkl}} + \varepsilon\_{\text{Normal}}^{\text{hkl}}\right) \tag{10}$$

where *Ehkl* and *vhkl* are the diffraction elastic constants (DECs) corresponding to the *hkl* reflection in each corresponding phase. In the present work, the DECs of Al2O<sup>3</sup> (*hkl*) and AT (*hkl*) were obtained using the program IsoDEC [35,36], following the Kröner model.
