*3.2. Elastic Modulus of UO2*

After the calculation of elastic constants, the bulk modulus, Young's modulus, and shear modulus of perfect UO2 are initially calculated from three independent elastic constants (*C*11, *C*12, and *C*44) using Equations (2), (5) and (6). Figure 4 depicts the dependence of the bulk modulus of perfect UO2 on temperature calculations of systems. For comparison, the experimental results from Belle et al. [33], the MD derived bulk modulus from Basak et al. [23], and ab initio data calculated by Wang et al. [26] are also included in this figure. It is clear that the present study has similar results to those obtained by previous MD calculations and experiments but lower than those from the VASP calculation. Figure 4 also indicates that the bulk modulus of perfect UO2 derived in this study decreases with an increase in temperature, which has been confirmed in the previous study by Dorado et al. [34].

**Figure 4.** Variation of the bulk modulus of pure UO2 versus temperature. For comparison, the experiment studies by Belle et al. [33], the Vasp data by Wang et al. [26] and the MD results by Basak et al. [23] are included.

Figure 5 shows the bulk modulus of the damaged UO2 at different temperatures as a function of Frenkel defect concentration (dash) and antisite defect concentration (solid). Figure 5 shows that both a Frenkel defect and an antisite defect could considerably decrease the bulk modulus of UO2, showing a linear decreasing dependence on temperature from 600 to 1500 K. With an increase in defect concentration, the elastic modulus also decreases accordingly, as shown by the figure. The extent of reduction in the elastic modulus for the system containing defects becomes smaller with increasing temperature and defect concentration. In addition, Figure 5 demonstrates that Frenkel defects increase the bulk modulus to a larger extent compared to that induced by antisite defects.

Figure 6 presents the effects of Frenkel (dash) and antisite defects (solid) on the shear modulus (*G*) of UO2. Firstly, for the concentration of defects investigated in this work, the shear modulus decreases with an increase in temperature. The extent of reduction in the shear modulus decreases with increases in temperature from 600 to 1500 K. Similar to the effects on the bulk modulus shown in Figure 5, it can be seen from Figure 6 that the increase of defect concentration could significantly reduce the shear modulus of UO2. However, the extent of reduction in *G* resulting from Frenkel defects is larger than that observed for antisite defects. For example, for 5% Frenkel and antisite defects there is a maximum of 20% and 17% reduction in *G* at all temperatures, respectively.

**Figure 5.** Variation of the bulk modulus of UO2 containing different concentrations of Frenkel (dash) and antisite defects (solid) versus temperature. The fitted lines are also included in the figure.

**Figure 6.** Variation of the shear modulus of UO2 containing different concentrations of Frenkel (dash) and antisite (solid) defects versus temperature. The fitted lines are also included in the figure.

Young's modulus of UO2 containing different concentrations of Frenkel (dash) and antisite defects (solid) as the function of the temperature is plotted in Figure 7. From this figure, it is clear that Young's modulus of uranium dioxide decreases linearly with the increase in temperature for Frenkel and antisite defects within the given concentrations. This result is similar to that reported by Jelea [13] et al. who observed that Young's modulus for damaged UO2 with different percentages of porosity linearly decreases with increases in temperature. Similar to the results of the bulk modulus and the shear modulus, Young's modulus also decreases with increasing temperature and concentration of defects. The relative change in *Y* due to Frenkel defects is larger than that observed for antisite defects. Within the given temperatures, a maximum of 20% reduction in *Y* is observed for UO2 systems containing 1%, 2%, and 5% Frenkel defects. In contrast, for antisite defects with the same concentrations, there is a maximum of 16% reduction in *Y* at all temperatures.

**Figure 7.** Variation of Young's modulus of UO2 containing different concentrations of Frenkel (dash) and antisite (solid) defects versus temperature. The fitted lines are also included in the figure.
