## 3 Simulation and Discussion
Using the fracture strength and elastic modulus of the silica fiber at room temperature [26; 30], the load-displacement curves from the FE model were simulated and are plotted in Figure 8. Based on these simulated load-displacement curves (Figure 8) and Equation (1), the simulated fracture toughness \(J_{C}\) at room temperature was calculated, and the observations showed that the simulated fracture toughness \(J_{C}\) of this 3D RF material, with ratios of 0.4 and 0.5, fit well with the experimental results (Figure 9) at 299 K. This observation demonstrates that the developed FE model is useful. In addition, based on the principal stress contours of the FE model as shown in Figure 10 (tensile displacement is 0.4 mm), we found that the maximum principal stress appeared on the fiber surface (Figure 10b), and that the fiber bears a bending deformation (Figure 10b). Our earlier work also revealed that the RF materials fracture through the breaking of the fibers, while almost all of the bonding between the fibers was kept intact when the RF material performed tensile displacement [31]. Figure 8: Simulated load–displacement curves from the FE model at room temperature (299 K), where the ratio was 0.5 from crack length to width. We showed in our previous study that the elastic modulus of the fibers continuously decreases from 299 to 1273 K [31]. However, the fracture strength of the fibers is hard to determine at elevated temperatures. Using this FE model and the elastic modulus of fibers, we predicted the fracture strength at elevated temperatures based on the predicting process in Figure S3 of the Supplementary Material. Additionally, the predicted tensile strength is plotted in Figure 11. Clearly, the fiber strength almost kept constant from 299 to 1073 K (\(\sim\)3600 MPa), and exhibited a sharp increasing trend as the temperature increased from 1073 to 1273 K (from \(\sim\)3600 to \(\sim\)5500 MPa). However, the elastic modulus exhibited a continuous decreasing trend from 299 to 1273 K (from \(\sim\)78 to \(\sim\)52 GPa). As shown in Figure 10, the simulated curves of the fracture toughness \(J_{C}\) vs. the temperature fitted well with the experimental observations. In fact, such a good fit cannot be obtained when the strength of the silica fibers, as well as the bonding between the fibers, continuously decreases with the increase in temperature. At an elevated temperature, the fibers, and the bonding between the fibers, usually softened and the crack tip of the CT specimen blunted. Therefore, more work is needed to stretch these fibers in this 3D RF material. This phenomenon can also be verified in the typical load-displacement curves from the CT specimen experiments at elevated temperatures (Figure 3). Figure 9: Experimental and simulated fracture toughness \(J_{C}\) of the CT specimen with crack length-to-width ratios of (**a**) 0.4 and (**b**) 0.5. Figure 11: Elastic modulus [26; 30] and the predicted elastic strength of the silica fibers and the bonding between the fibers at various temperatures. Figure 10: Principal stress contours of (**a**) the FE model with a ratio of 0.5 (the tensile displacement was 0.4 mm), (**b**) the stress contours of the sliced FE model from the red rectangle in (**a**), and (**c**) the stress contours of the local structure from the red arrow in (**b**). Moreover, from room temperature to 1273 K, the strength of the fused-silica materials increased from 49.8 \(\pm\) 3 MPa to 76.9 \(\pm\) 17 MPa [32]. It increased by approximately 154%. Such an increase in strength at a high temperature has been found to be attributed to the lifted atom mobility, resulting in a limited amount of crack-tip blunting [32; 33]. In this work, the fracture strength of the silica fibers increased from 3600 [31] to ~5500 MPa between temperatures of 299 and 1273 K; this increased rate of fracture strength was consistent with the fused-silica materials. In addition, the 3D RF material was sintered using a few mullite fibers and substantial amount of high purity silica fibers in this study.