## 1 Introduction  
The 3-dimensional (3D) random fibrous (RF) material, sintered by inorganic fibers, is one kind of composite material that is notable for its low weight, high porosity, high specific surface area, remarkable insulation properties, etc. Due to these individual performances, the 3D RF materials possess a wide application in engineering fields, such as a space shuttle's thermal insulation, building and filtration [1; 2; 3; 4; 5]. However, the complicated mechanical-thermal coupled service environments of the space shuttle require that the 3D RF materials possess prominent thermal insulation properties and a good mechanical performance. Moreover, this kind of material is promising, as these materials have, relatively, rather high mechanical features but quite a low crack resistance property at the same time. In addition, the crack resistance is critical for ceramic materials that are acting under extreme thermal and mechanical loads, for example, some structures whose brittle fracture is unendurable, even under random loads [6]. Fracture toughness, as a generic term of the material property for crack resistance evaluations, can be directly applicable to fracture control [7]. Moreover, fracture toughness is also a coarse-graining property--when a crack extension crosses the specimen, it supplies a perceived measurement at a certain dimensional scale of the energy dissipation [8]. In addition, some works focus on the temperature effect on the fracture toughness of brittle materials [9; 10]. For example, the micro-scale experiment results reveal that the initiation of small-scale plasticity plays an important role in the brittle-to-ductile transition of the material, such as miniaturized Si, at an elevated temperature. However, the measured fracture toughness \(J_{\text{C}}\) of some metals in the ductile-to-brittle transition temperature region reveals that there are test specimen size effects, even if the specimen is a standardized specimen [11; 12]. In addition, the fracture toughness of composite materials can also be estimated using the J-integral [13; 14], which was devised in 1968 by [PERSON] [15] to calculate the strain energy release rate and the energy per unit fracture surface area of the materials. Up to now, several methods have been developed for estimating the fracture toughness or slow stable crack growth of inelastic or nonlinear materials and metallic materials. For example, the most commercial FE analysis software can be applied to calculate the J-integral at a 2D crack tip or along a 3D crack front for actual structures. Some approaches are used to simulate the damage properties of composite materials, such as using cohesive elements [16], smeared crack models [17] or X-FEM [18]. Each of these requires the fracture toughness as a judging criterion for the failure mode analysis in the simulating process. Furthermore, a representative volume element (RVE), assumed in the micromechanics theory, is associated with the continuum damage mechanics to describe the development of the damage properties. Additionally, thermodynamic models are established to describe the progressive failure performances and to account for the stiffness degradation of the composite materials [19; 20; 21; 22; 23]. Although much effort has been invested in the fracture toughness of metals and composite materials, the fracture toughness \(J_{\text{C}}\) model for the 3D RF material has not been well developed, and the temperature dependence of the fracture mechanism is not revealed either. In the previous work, we studied the fracture toughness (\(K_{\text{IC}}\)) of 3D RF materials from 299 to 1273 K by experiments and the FEM, just based on the critical load \(P_{Q}\)[24] in the load displacement curves of compact tension (CT) experiments. Furthermore, we have also uncovered the fracture mechanism of these 3D RF materials at the crack tip [25]. However, the simulated fracture toughness (\(K_{\text{IC}}\)) only fits well with the experimental observations at a low temperature.