*3.3. Reduction of Elastic Modulus of UO2 by Frenkel Defects and Antisites*

The effects of Frenkel defects and antisites on elastic modulus are reported in detail in Figures 8–10. In each of these figures, (a) and (b) represent the influence of Frenkel defects and antisite defects on the elastic modulus, respectively. The percentage of reduction (*R*) in the elastic modulus as the function of fractional point defects for all temperatures is plotted. The percentage of reduction of the elastic modulus is calculated using the following Equation (7).

$$R = (M\_{\mathbb{P}} - M) / M\_{\mathbb{P}} \tag{7}$$

where *M*<sup>p</sup> and *M* represent the elastic modulus of a perfect and defective UO2, respectively.

**Figure 8.** *R*<sup>B</sup> as a function of the concentration of Frenkel defects (**a**) and antisites (**b**) at different temperatures.

**Figure 9.** *R*<sup>G</sup> as a function of the concentration of Frenkel defects (**a**) and antisites (**b**) at different temperatures.

**Figure 10.** *R*<sup>Y</sup> as a function of the concentration of Frenkel defects (**a**) and antisites (**b**) for different temperatures.

Figures 8a, 9a and 10a illustrate that in low concentration ranges the difference among the percentages of reduction in the elastic modulus for different temperatures is small. However, this difference is more evident at higher defect concentrations. In addition, this is not as significant for antisite defects as it is for Frenkel defects. The *R* of the elastic modulus due to antisite defects is smaller than that by Frenkel defects. For 5% defects, the degradation of the bulk modulus, shear modulus, and Young's modulus by Frenkel defects is higher than that of antisite defects by 3.4, 3.9, and 2.9%. Therefore, by comparing Figures 8–10, it can be observed that the percentage of reduction in the elastic modulus by Frenkel and antisite defects follows the trend *R*<sup>f</sup> > *R*<sup>a</sup> for all studied defect concentrations.

#### **4. Conclusions**

In order to characterize the effect of irradiation on the performance of the nuclear fuel, it is necessary to investigate how the irradiation defects affect the thermal-mechanical property of nuclear fuel. In this study, the impact of Frenkel defects and antisites on thermal expansion and elastic constants has been examined in UO2 via the molecular dynamics method in the temperature range of 600 to 1500 K. The results indicate that both Frenkel defects and antisite defects reduce the thermal expansion coefficient. However, the reduction in the thermal expansion coefficient due to antisite defects is larger than that observed for Frenkel defects. For the elastic modulus, the calculated bulk, shear, and Young's modulus of the pure UO2 are in agreement with the experimental values. Furthermore, the present results indicate that both Frenkel defects and antisite defects reduce the elastic modulus at all temperatures. The degree of reduction in the elastic modulus increases with increasing concentrations of defect. In addition, the percentage of reduction in the elastic modulus due to Frenkel and antisite defects follows the trend *R*<sup>f</sup> > *R*<sup>a</sup> at all studied defect concentrations. All these calculated values can be used to predict the performance of UO2 under irradiation used in the nuclear reactor environment.

**Author Contributions:** Conceptualization, Z.W. and N.G.; methodology, Z.W.; software, M.Y.; validation, C.Y., X.L. and Z.W.; formal analysis, Z.W.; investigation, M.Y.; resources, N.G.; data curation, Z.W.; writing—original draft preparation, Z.W.; writing—review and editing, N.G.; visualization, M.Y. and L.D.; supervision, N.G. and Z.Y.; project administration, Z.Y.; funding acquisition, N.G. and X.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was financially supported by the National Natural Science Foundation of China (Project Nos. 12075141 and 12175125).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.
