3.4.2. Peak-Specific Residual Stresses

Peak-specific residual stresses were assessed by measuring the shifts in individual diffraction peaks in the TOF diffraction spectrum. Considering the peak intensity, as well as the non-overlapping and clear shape in the diffraction spectra, the peaks of Al2O<sup>3</sup> (i.e., (030), (116), (024), (113)) and AT (i.e., (243), (200), (043), and (113)) were selected for analysis, as shown in Figure 6. The d-spacing of each peak was obtained by Rietveld refinement. Peakspecific residual stresses for the selected peaks were calculated using Equations (9) and (10). The through-thickness peak-specific residual stress behaviors of selected reflections ysis, as shown in Figure 6. The d-spacing of each peak was obtained by Rietveld refinement. Peak-specific residual stresses for the selected peaks were calculated using Equations (9) and (10). The through-thickness peak-specific residual stress behaviors of selected reflections

as the non-overlapping and clear shape in the diffraction spectra, the peaks of Al2O3 (i.e., (030), (116), (024), (113)) and AT (i.e., (243), (200), (043), and (113)) were selected for anal-

of Al2O<sup>3</sup> and AT in all samples, measured in both the in-plane and normal directions, are plotted in Figures 8 and 9, respectively. of Al2O3 and AT in all samples, measured in both the in-plane and normal directions, are plotted in Figures 8 and 9, respectively.

*Materials* **2021**, *14*, 7624 18 of 24

**Figure 8.** Through-thickness residual stress profiles corresponding to alumina reflections (i.e., (116), (024), (113), and (030)) in Al2O3–Al2TiO5 composites (1450 °C in blue line and 1550 °C in red line) measured along inplane (solid line and solid symbol) and normal (dash line and hollow symbol) directions. **Figure 8.** Through-thickness residual stress profiles corresponding to alumina reflections (i.e., (116), (024), (113), and (030)) in Al2O3–Al2TiO<sup>5</sup> composites (1450 ◦C in blue line and 1550 ◦C in red line) measured along inplane (solid line and solid symbol) and normal (dash line and hollow symbol) directions.

**Figure 9.** Through-thickness residual stress profiles corresponding to AT reflections (i.e., (043), (113), (243), and (200)) in Al2O3–Al2TiO5 composites (1450 °C in blue line and 1550 °C in red line) measured along in-plane (solid line and solid symbol) and normal (dash line and hollow symbol) directions. **Figure 9.** Through-thickness residual stress profiles corresponding to AT reflections (i.e., (043), (113), (243), and (200)) in Al2O3–Al2TiO<sup>5</sup> composites (1450 ◦C in blue line and 1550 ◦C in red line) measured along in-plane (solid line and solid symbol) and normal (dash line and hollow symbol) directions.

As shown in Figures 8 and 9, the through-thickness residual stress profiles of all the selected reflections are virtually flat in both the in-plane and normal directions, confirming the occurrence of mean phase stress behaviors as previously discussed. Variations among peak-specific residual stresses are not remarkable between the in-plane and normal directions. The average peak-specific residual stress values of the selected peaks for both the Al2O3 and AT phases are summarized in Tables 4 and 5, respectively; these values were obtained regardless of the measurement directions. To clearly contrast the stresses of the Al2O3 and AT phases, their average mean phase stresses are also listed. As shown in Figures 8 and 9, the through-thickness residual stress profiles of all the selected reflections are virtually flat in both the in-plane and normal directions, confirming the occurrence of mean phase stress behaviors as previously discussed. Variations among peak-specific residual stresses are not remarkable between the in-plane and normal directions. The average peak-specific residual stress values of the selected peaks for both the Al2O<sup>3</sup> and AT phases are summarized in Tables 4 and 5, respectively; these values were obtained regardless of the measurement directions. To clearly contrast the stresses of the Al2O<sup>3</sup> and AT phases, their average mean phase stresses are also listed.

**Table 4.** Average residual stress values of selected peaks for Al2O3 compared with corresponding mean phase stresses. **Table 4.** Average residual stress values of selected peaks for Al2O<sup>3</sup> compared with corresponding mean phase stresses.


(1550) −176 ± 6 −283 ± 7 −125 ± 9 −195 ± 8 −215 ± 7

(1450) −194 ± 6 −230 ± 6 −196 ± 7 −207 ± 8 −14 ± 7

(1550) −183 ± 7 −219 ± 4 −195 ± 5 −201 ± 4 −208 ± 5

A-10AT

A-40AT

A-40AT


**Table 5.** Average residual stress values of selected peaks for AT phase compared with corresponding mean phase stresses.

For the same phase in the same sample, the obtained peak-specific residual stress values varied among different *hkl* reflections. Compression was developed in all the selected *hkl* reflections of the Al2O<sup>3</sup> phase in the studied A–AT composites. However, the AT phase underwent tension in the reflections of (043) and (113); compression occurred in (243) and (200). The values of the peak-specific residual stresses in each phase differed from their mean phase stresses. As the AT content increased from 10 to 40 vol.%, the variation ranges of residual stresses in different reflections became smaller for both the Al2O<sup>3</sup> and AT phases. However, the orders of peak-specific residual stress values in different *hkl* reflections were unchanged in each phase.

The effects of different sintering temperatures on peak-specific residual stresses were investigated. With a higher sintering temperature, the A-10AT composites exhibited reduced absolute values of compression in all selected *hkl* reflections of Al2O3; however, no distinct difference in peak-specific residual stresses of the AT phase was observed. Conversely, in the A-40AT composites, the peak-specific residual stresses in the Al2O<sup>3</sup> phase between A-40AT(1450) and A-40AT(1550) were similar, whereas the AT phase exhibited evident differences in peak-specific residual stresses between these two samples. With a higher sintering temperature in the A-40AT composites, the variation range of residual stresses in different *hkl* reflections were reduced in the AT phases.

The results of peak-specific residual stresses, calculated from the d-spacing of each reflection, can be explained by the lattice parameter values. For hexagonal α-Al2O3, the relationship between the d-spacing (*dhkl*) for a given *hkl* plane and the lattice parameters, *a* (*a* = *b*) and *c*, is written as follows [50]:

$$d\_{hkl} = \frac{1}{\sqrt{\frac{4}{3a^2}(h^2 + k^2 + hk) + \frac{l^2}{c^2}}} \tag{12}$$

According to the obtained lattice parameters listed in Table 3, the *a* and *c* axes of α-Al2O<sup>3</sup> shrunk in all the studied A–AT composites when compared with those of the Al2O<sup>3</sup> reference powder. Thus, according to Equation (12), for a given lattice plane *hkl* of Al2O3, the d-spacing decreased, and compression developed in all the selected reflections in the Al2O<sup>3</sup> phase in the A–AT composites.

As the AT content increased from 10 to 40 vol.%, the *a* axis of Al2O<sup>3</sup> slightly expanded, but the *c* axis shrunk. This led to peak-specific residual stresses in some *hkl* reflections and a certain reduction in the Al2O<sup>3</sup> phase. For example, for *hk*0 planes in the Al2O<sup>3</sup> phase in the A-40AT composites, with a slightly increased *a* value, *dhk*<sup>0</sup> increased, and the absolute values of the compressive peak-specific residual stresses decreased compared with those in the A-10AT composites. This is consistent with the measured residual stresses of A(030) in the A–AT composites, as listed in Table 4.

Compared with the A-10AT(1450) composites, the A-10AT(1550) composites showed slight increases in the *a* and *c* axes of α-Al2O3. Thus, with a higher sintering temperature value, the d-spacing of each Al2O<sup>3</sup> reflection slightly increased, and the absolute values of compressive peak-specific residual stresses decreased. In the A-40AT composites, the difference in the lattice parameters of Al2O<sup>3</sup> was not distinct at different sintering temperatures; thus, no remarkable difference was observed in the peak-specific residual stresses of Al2O<sup>3</sup> in the two samples.

For orthorhombic AT, with the lattice parameters *a*, *b*, and *c*, the interplanar spacing (*dhkl*) is given by the following:

$$d\_{hkl} = \frac{1}{\sqrt{\frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2}}}\tag{13}$$

As summarized in Table 3, the *a* axis of the AT phase shrunk, and the *b* and *c* axes expanded in all the studied A–AT composites compared with those of the AT reference powder. Thus, for a specific *hkl* plane of AT, the d-spacing may decrease or increase according to Equation (13). This leads to peak-specific residual stresses, either compression or tension in different selected reflections in AT. For example, for the 0*kl* reflections, *d*0*kl* increased, and tension developed. For the *h*00 reflections of AT, *dh*<sup>00</sup> decreased, and compression developed.

As the AT content increased from 10 to 40 vol.%, the *a* axis of the AT phase expanded, and the *b* and *c* axes shrunk. Thus, for the 0*kl* reflections in AT, the d-spacing decreased, and tension decreased. For the *h*00 reflections in AT, *dh*<sup>00</sup> increased, and the absolute values of compression decreased. These are all in accordance with the measured peak-specific residual stress behaviors of AT(043) and (200), as summarized in Table 5.

With regard to the effect of different sintering temperatures, owing to the similar lattice parameters of AT in the A-10AT(1450) and A-10AT(1550) samples, the peak-specific residual stresses for each selected AT reflection were similar in both samples. In the A-40AT composites, as the sintering temperature increased from 1450 to 1550 ◦C, the *a* axis of the AT phase expanded, and the *b* and *c* axes shrunk. Consequently, for AT(430) in A-40AT(1550), *d*<sup>043</sup> decreased, and the tension decreased with respect to that in A-40AT(1450). For AT(200) in A-40AT(1550), *d*<sup>200</sup> increased; consequently, the absolute compression value decreased with respect to that in A-40AT(1450).

Owing to the strong anisotropy of thermal expansion in the AT phase, with higher AT content and sintering temperature, more microcracks are generated in the A–AT composites. This leads to the reduced connectivity of the material in the composites; thus, the restraints in the material are weakened. Hence, the anisotropy of peak-specific residual stresses in both Al2O<sup>3</sup> and AT phases was weaker in A-40AT than in the A-10AT composites, and the anisotropy of the peak-specific residual stresses of the AT phase was weaker in A-40AT(1550) than in A-40AT(1450).

As mentioned above, it is well accepted that peak-specific residual stress behaviors are mainly determined by the crystal structure of each phase; this is closely related to the CTE anisotropy in each phase and the microstructure of materials. On the one hand, owing to the anisotropy of thermal expansions and elastic properties in each *hkl* direction, for the same sample and phase, the residual stress values obtained vary from different reflections. The sign and magnitude of residual stress values considerably depend on the reflection used for analysis in the diffraction method. This is an important concern in residual stress measurements and analysis using the diffraction method with single-peak reflections in complex materials. On the other hand, with looser microstructures, such as microcracking, the anisotropy of peak-specific residual stresses in each phase was distinctly weakened. Deriving reliable properties is beneficial owing to lower internal stresses. However, a looser structure may limit the strength of materials. Thus, to balance the weakening of the anisotropy of residual stresses and improve mechanical properties, optimizing and controlling the microstructure of materials are crucial.
