## 3 Simulation and Discussion  
In order to evaluate the blunting deformation and fracture toughness \(J_{C}\) of the RF material at an elevated temperature, a 3D FE model, with the morphology characteristics of the RF material and a crack, was developed. This developed FE model was a scaled model with the CT specimen shape. The width W1 of the FE model was made a proportion 1250/21 of the CT specimen size W (Figure 6). The modeling and calculating details are supplied in the Supplementary Material. Furthermore, the whole simulation process was performed in the ANSYS software, and Beam 188 element was applied to study the fibers and the bonding between the fibers. In the CT specimen, there were two holes--this  
Figure 5: (**a**) The fracture toughness \(J_{C}\) of the CT specimens at various temperatures, with crack length – to – width ratios of 0.4 and 0.5, respectively. The crack tip of the specimen (a / W = 0.4) at (**b**) 299 K and (**c**) 1123 K after testing. The red arrows denote the direction of the crack propagation. (**d**) An image of a CT specimen tested in the furnace. was a limitation in our modeling. In order to overcome this limitation, firstly, we built the FE model based on the set porosity. Then, we used the Boolean operation in ANSYS software to cut two holes in the FE model. In this process, we avoided the Boolean operation affecting the porosity of the FE model. In addition, there were about 39,000 nodes and 41,000 elements in the FE model. The constraint equations between the two RVE (Representative Volume Element) models, the load position and the fixed support are marked in the FE model, as shown in Figure 7. Moreover, based on our experiments of the silica fiber bundle at elevated temperature, we found that the silica fiber bundle exhibited an elastic-fragile behavior. However, the viscous flow deformation temperature of fused SiO\({}_{2}\) was about 800 \({}^{\circ}\)C. So, we inferred that the silica fiber exhibited elastic-viscoelastic behavior above the critical temperatures. Moreover, the elementary composition of the bonds was close to the fibers. They should have exhibited a similar elastic-viscoelastic behavior. Therefore, we considered the bond as a short silica fiber in the simulation. However, we assumed that the silica fiber exhibited an elastic-fragile behavior in the simulation. Then, we were able to use the elastic modulus and the fracture strength of the silica fibers to evaluate the mechanical properties of the 3D RF material at elevated temperatures. Thus, we could also overcome the effect of time in the constitutive equation of the viscous flow deformation in the silica fiber. Figure 6: The FE model of the CT specimen (the width \(W_{1}\) is \(1.4L\), \(L\) is the average fiber length (0.6 mm), the thickness of the FE model \(B_{1}\) is \(W_{1}\)/2). Figure 7: Constraint equations, load and fixed support shown in the FE model. In addition, all of the silica fibers and bonds in the FE model were considered to be isotropic materials, and then there were only two independent material parameters in constitutive relationships. Moreover, the fibers were all fragile, thus the maximum principal stress criterion was used to predict the onset of their damage. In the analysis, the temperature-dependent elastic modulus and fracture strength were just considered, while the Poisson's ratio was independent of the temperatures. By doing so, we were able to simulate the mechanical behavior of the 3D RF materials at an elevated temperature. In our previous study, the fracture strength and elastic modulus of the silica fibers were considered as a decreasing variation trend with the temperature. As a result, the simulated fracture toughness showed a decreased variation trend and fit well with the experimental fracture toughness before 773 K [25]. This difference has been attributed to the viscous flow deformation [29] of fibers and bonds between fibers.