## 4 Numerical Examples
### Three-Point Bending Test
Additionally, the boundary conditions in the experimental model may not be as ideal as those in numerical simulations, and the heterogeneity of the brittle solid sheet is not taken into account. Figure 8: Geometry and boundary conditions for the numerical model. Figures 11 and 12 illustrate the distribution of the maximum principal stress and temperature of the brittle solid sheet with varying initial temperatures at 100 ms, respectively. Figure 10: Comparison of experimental and numerical results for different types of cracks. Figure 9: Comparison of numerical and experimental results [48] of the final thermal shock crack patterns. The maximum principal stress is concentrated at the crack tips. This is due to the significant shrinkage deformation caused by the sharp temperature drop of the outer boundary of the brittle solid sheet during thermal shock, which induces large tensile stresses. Interestingly, in the temperature field distribution (see Figure 12), the temperature exhibits discontinuity at both ends of the crack in the brittle solid sheet. This is because the crack impedes heat conduction, meaning no heat passes through the cracks. A brittle solid sample with an initial temperature of 600 \({}^{\circ}\)C is chosen for an in-depth analysis of the crack propagation mechanism under thermal shock. Figures 13 and 14 depict the internal temperature and maximum principal stress distribution of the 600 \({}^{\circ}\)C sample at different times. At 3 ms, thermal shock is in its early stage. A steep temperature gradient forms at the boundary, causing large tensile stresses there. Numerous short cracks start to form at the left and upper boundaries. Upon crack initiation, the stresses at the crack sites are relieved and subsequently redistributed throughout the sheet. Between 3 ms and 50 ms, as the cryogenic zone propagates inward, additional regions within the specimen attain the tensile strength threshold defined by the maximum principal stress. This makes existing cracks grow or new ones form. Each time new cracks appear, stresses readjust until thermal stresses from the temperature gradient can no longer drive crack growth. Eventually, cracks compete to form different length morphologies (see Figure 15). The initial abundance of short cracks and subsequent crack initiation and propagation reflect the system's rapid energy minimization. This aligns with [PERSON] fracture theory and well explains the fracture variational phase-field theory. Figure 11: Maximum principal stress distribution at different initial temperatures. Figure 12: Temperature distribution at different initial temperatures. Thermal shock stimulation is utilized to create fracture networks in reservoir rocks, enhancing production. The heat transfer coefficient is a key parameter for evaluating the effectiveness of this process. To elucidate the influence of the heat-transfer coefficient on thermally-shock-driven fracture in brittle solids, experiments were performed on specimens initially equilibrated at 600 \({}^{\circ}\)C, employing heat-transfer coefficients of 50,000, 100,000, and 500,000 W/(m\({}^{2}\cdot\)K). Figure 16 demonstrates that an increase in the heat-transfer coefficient yields a higher crack count and a corresponding reduction in crack spacing. Based on prior crack length classification, the number of long and medium-length cracks is largely unaffected by the heat transfer coefficient. However, short cracks rise sharply with increasing heat transfer coefficient, driving up the total crack count (Figure 17). This is because a higher heat transfer coefficient speeds up the boundary cooling rate and intensifies the temperature gradient. The system then releases energy rapidly through numerous short cracks to minimize total energy. In practical thermal shock stimulation, along with fluid medium temperature control, enhancing fluid properties (e.g., type and flow rate) and fluid-reservoir rock contact area can increase the heat transfer coefficient. This improves reservoir permeability and boosts geothermal energy recovery.