3.4.1. Mean Phase Stresses

Mean phase stress, the average stress in each phase over several randomly oriented grains within the gauge volume area in the TOF measurement, was calculated using the change in the lattice parameters of the Al2O<sup>3</sup> and AT phases, as given by Equations (1)–(6), respectively. The through-thickness mean phase stress profiles measured in the in-plane and normal directions are depicted in Figure 7 for the A–AT bulk samples with different AT additions and sintering temperatures (1450 and 1550 ◦C). Note that error bars corresponding to the statistical uncertainties of the determined lattice parameters are smaller than the size of the symbols employed in the presented graphs.

**Figure 7.** Mean phase stress profiles for Al2O3 (Δ) and Al2TiO5 (□) phases; through-thickness of Al2O3–Al2TiO5 bulk samples as a function of Al2TiO5 contents and sintering temperatures (1450 °C in blue line and 1550 °C in red line); values of + ்் are presented ( ) without a line. (**a**) A-10AT composites and (**b**) A-40AT composites. **Figure 7.** Mean phase stress profiles for Al2O<sup>3</sup> (∆) and Al2TiO<sup>5</sup> () phases; through-thickness ofAl2O3–Al2TiO<sup>5</sup> bulk samples as a function of Al2TiO<sup>5</sup> contents and sintering temperatures (1450 ◦C in blue line and 1550 ◦C in red line); values of *fAσ<sup>A</sup>* + *fATσAT* are presented ( **Figure 7.** Mean phase stress profiles for Al2O3 (Δ) and Al2TiO5 (□) phases; through-thickness of Al2O3**–**Al2TiO5 bulk samples as a function of Al2TiO5 contents and sintering temperatures (1450 °C in blue line and 1550 °C in red line); values of + are presented ( ) without a line. (**a**) A-10AT composites and (**b**) A-40AT composites. ) without a line. (**a**) A-10AT composites and (**b**) A-40AT composites.

As shown in Figure 7, the mean phase stress behaviors between the normal and inplane directions are similar in all cases. The through-thickness stress profiles of both AT and Al2O3 phases are virtually flat, with residual tensile stresses in AT particulates and compressive stresses in the Al2O3 matrix. The high tensile stresses in AT were approximately 500–610 MPa for the A-10AT composites irrespective of the sintering temperature. It rapidly decreased as the AT content increased to approximately 80–180 MPa in the A-40AT composites. As the AT vol.% changed, the compressive stresses in the Al2O3 did not vary and remained at approximately −200 ± 30 MPa. As shown in Figure 7, the mean phase stress behaviors between the normal and inplane directions are similar in all cases. The through-thickness stress profiles of both AT and Al2O3 phases are virtually flat, with residual tensile stresses in AT particulates and compressive stresses in the Al2O3 matrix. The high tensile stresses in AT were approximately 500**–**610 MPa for the A-10AT composites irrespective of the sintering temperature. It rapidly decreased as the AT content increased to approximately 80**–**180 MPa in the A-40AT composites. As the AT vol.% changed, the compressive stresses in the Al2O3 did not vary and remained at approximately −200 ± 30 MPa. As shown in Figure 7, the mean phase stress behaviors between the normal and in-plane directions are similar in all cases. The through-thickness stress profiles of both AT and Al2O<sup>3</sup> phases are virtually flat, with residual tensile stresses in AT particulates and compressive stresses in the Al2O<sup>3</sup> matrix. The high tensile stresses in AT were approximately 500–610 MPa for the A-10AT composites irrespective of the sintering temperature. It rapidly decreased as the AT content increased to approximately 80–180 MPa in the A-40AT composites. As the AT vol.% changed, the compressive stresses in the Al2O<sup>3</sup> did not vary and remained at approximately −200 ± 30 MPa.

In addition, the residual stress behaviors according to different sintering treatments were studied. Regarding A–AT ceramics with the same AT addition, as the sintering temperature increased from 1450 to 1550 °C, tension in the AT phase decreased, and absolute compression values in the Al2O3 matrix slightly decreased in the A-10AT composites; however, no distinct variations were found in the A-40AT composites. In addition, the residual stress behaviors according to different sintering treatments were studied. Regarding A**–**AT ceramics with the same AT addition, as the sintering temperature increased from 1450 to 1550 °C, tension in the AT phase decreased, and absolute compression values in the Al2O3 matrix slightly decreased in the A-10AT composites; however, no distinct variations were found in the A-40AT composites. In addition, the residual stress behaviors according to different sintering treatments were studied. Regarding A–AT ceramics with the same AT addition, as the sintering temperature increased from 1450 to 1550 ◦C, tension in the AT phase decreased, and absolute compression values in the Al2O<sup>3</sup> matrix slightly decreased in the A-10AT composites; however, no distinct variations were found in the A-40AT composites.

The residual thermal stresses in a particulate-reinforced composite are known to be caused by the elastic deformations of the matrix and particulates under uniform temperature change [45]. This is mainly due to the mismatch of thermal expansions and elastic constants between the matrix and particulates. For A–AT composites, the average crystallographic thermal expansion coefficient for the Al2O3 matrix is smaller than that for the AT inclusion [13,46,47]. Therefore, during fabrication, which is subjected to cooling from The residual thermal stresses in a particulate-reinforced composite are known to be caused by the elastic deformations of the matrix and particulates under uniform temperature change [45]. This is mainly due to the mismatch of thermal expansions and elastic constants between the matrix and particulates. For A**–**AT composites, the average crystallographic thermal expansion coefficient for the Al2O3 matrix is smaller than that for the AT inclusion [13,46,47]. Therefore, during fabrication, which is subjected to cooling The residual thermal stresses in a particulate-reinforced composite are known to be caused by the elastic deformations of the matrix and particulates under uniform temperature change [45]. This is mainly due to the mismatch of thermal expansions and elasticconstants between the matrix and particulates. For A–AT composites, the average crystallographic thermal expansion coefficient for the Al2O<sup>3</sup> matrix is smaller than that for the AT inclusion [13,46,47]. Therefore, during fabrication, which is subjected to cooling from the

maximum sintering temperature to room temperature, high compressive residual thermal stresses were induced in the Al2O<sup>3</sup> matrix and tension in the AT inclusion.

Microstructural factors, such as particle volume fraction, size, shape, and microcracking, are known to also affect the magnitude and distribution of residual stresses. Considering the microstructure and grain size of AT and Al2O<sup>3</sup> in A–AT samples with high AT contents, the grain size of AT was observed to increase. For a single AT particle, the surrounding misfit effects weakened because of the reduction in the contact area with the Al2O<sup>3</sup> matrix and the increase in the contact area among AT particulates. The tension produced in AT due to the CTE misfit with the Al2O<sup>3</sup> matrix correspondingly weakened. Furthermore, spontaneous AT particle microcracking occurred as the AT content increased to 40 vol.%. The occurrence of microcracking can relieve the stress energy in the AT phase. Both effects contributed to the remarkable decrease in tensile stresses in the AT phase of the A–AT composites with increasing AT content.

No significant change was detected in the stress value of the Al2O<sup>3</sup> matrix as the addition of second-phase AT increased. This was inconsistent with the trend in other alumina-based ceramic composites reinforced by the second phase with higher thermal expansion, showing an increase in the absolute value of compression in the Al2O<sup>3</sup> matrix [33,48]. This could be attributed to the special microcracking characteristics of AT particulates in A–AT ceramics. As AT vol.% increased, more microcracks were generated in AT and propagated along the boundaries of Al2O<sup>3</sup> and AT. Microcracking benefited the absorption of stress energies during the expansion and shrinkage of materials during fabrication. Consequently, the stress state of the Al2O<sup>3</sup> matrix in the A–AT composites was not significantly affected by the increase in AT content. This effect was also demonstrated by the thermal expansion curve of A–AT composites reported in a previous study [49], showing significant thermal expansion hysteresis in A-40AT compared with the A-10AT composites.

After sintering, temperature changes during cooling can lead to a higher CTE misfit strain value, thus increasing stresses. However, our measurement showed lower tension in the AT phase in the A–AT composites with higher sintering temperatures. Microcracking was presumed to be responsible for this inconsistency. Based on Figure 3 and Table 2, at higher sintering temperatures, the grain sizes of both AT and Al2O<sup>3</sup> increase, resulting in further microcracking as a higher population of AT grains surpasses its critical size. This contributes to the release of thermal stress energy from samples with higher sintering temperatures; consequently, the tension in the AT phase decreases.

Moreover, according to the derived mean phase stresses of the Al2O<sup>3</sup> matrix and AT particles derived by neutron diffraction, the macro-residual stresses, *σbulk*, of the A–AT bulk samples were calculated using the following equation:

$$
\sigma\_{\text{bulk}} = f\_A \sigma\_A + f\_{\text{AT}} \sigma\_{\text{AT}} \tag{11}
$$

where *f<sup>A</sup>* and *fAT* are the volume fractions of Al2O<sup>3</sup> and AT, respectively; and *σ<sup>A</sup>* and *σAT* are the mean phase stresses of Al2O<sup>3</sup> and AT, respectively.

The through-thickness values, *σbulk*, of the A–AT bulk samples are plotted in Figure 7. Macro-residual stresses, *σbulk*, were initially assumed negligible in all the A–AT bulk samples because no pressure was applied during the sintering process. However, according to the results, the *σbulk* values were not zero but compressive in the A–AT samples. This could be explained by considering the unknown stress states near the surface areas. As mentioned, strain scanning using neutron diffraction measurements was implemented 2 mm away from sample surfaces. The stress state near the surface was unknown because it could not be determined by neutron diffraction. Based on the hydrostatic assumption, tensile stress was anticipated around the surface area of the A–AT bulk.
