## 5 Numerical Examples
### The 2D Quenching Test  
In this example, the complex crack patterns arising from thermal shocks are investigated by the proposed NOSB-PD model. In previous experiments, the ceramic slabs were heated to an initial temperature \(T_{0}\), which ranged from 300 \({}^{\circ}\)C to 600 \({}^{\circ}\)C, and then the heated specimen was dropped into a water bath of \(T_{\mathrm{ref}}=20\)\({}^{\circ}\)C by free-fall [3]. The 2D model is considered and the geometry and boundaries are given in Figure 12. To save computational cost, a one-quarter model (lower left part) replaces the full-size model. The initial temperature is set to be 500 \({}^{\circ}\)C, and the convective heat transfer coefficient \(h\) between solid and water is set to be \(700,000\) W/ \(\left(\mathrm{m}^{2}\cdot{}^{\circ}\mathrm{C}\right)\)[37]. Moreover, the horizon \(\delta\) is set to be 0.25 mm, and \(l\) is set to be 0.083 mm. In addition, the mechanical properties of the ceramic slab given in Table 4 are approximately unchanged in the range 200 \({}^{\circ}\)C to 600 \({}^{\circ}\)C. The time increment is chosen as \(\Delta t=0.005\)\(\upmu\)s. The total computational time is 200 ms.  
Figure 11: Contour plots at different times: (**a**) damage \(D\) and (**b**) fracture level \(d\) in case i, (**c**) damage \(D\) and (**d**) fracture level \(d\) in case ii. Figure 12: The geometry and boundaries of the specimen. Figure 13 shows contour plots of the temperature, damage, and crack growth paths compared with the experiment result at the final time. It can be seen that the temperature undergoes significant cooling. The accompany temperature gradient results in damage to the surfaces of the specimen. With the change in damage state, cracks further occur. The results also show that the thermal cracks are approximately parallel and equally spaced of similar length, which demonstrates that the numerical crack pattern is quite similar to the experimental one [3]. Figure 14 also shows the evolution of temperature fields and crack growth paths during the simulation. At the early stage of thermal shock, cracks are initiated on the left and bottom surfaces at \(t=10\) ms. In the rest of the surface area, a series of small cracks almost at equal spacing are generated. These cracks occur simultaneously due to a huge temperature gradient. Subsequently, only a few cracks continue to propagate upwards [see Figure 14, (\(t=30\) ms)]. During the propagating process, some cracks grow faster and become long cracks, while others grow in a much slower way and remain short in the end [see Figure 14, (\(t=100-200\) ms)]. It can be found that the periodical and hierarchical characteristics of crack spacing and length that are formed in the numerical simulations are in good agreement with the experimental observations. Moreover, the significant temperature jumping across all the bent thermal cracks can also be captured in the simulation using the proposed model. In addition, another case with initial temperature \(300\)\({}^{\circ}\)C is considered to further demonstrate the ability of the proposed NOSB-PD model in simulating crack propagation under different thermal shocks. Figure 15 illustrates the final crack patterns of the numerical and experimental results in the quenching tests. It can be observed that the number of cracks increases with initial temperature. The result, considering an initial temperature of \(300\)\({}^{\circ}\)C, also shows the absence of the distinct hierarchy of crack lengths, which can be attributed to the effect of the fracture strain \(\bar{\epsilon}_{\rm f}^{\rm p}\). In fact, a smaller \(\bar{\epsilon}_{\rm f}^{\rm p}\) will make the material more brittle in terms of its fracture property. These results imply the effectiveness of the proposed NOSB-PD model for simulating and predicting thermodynamic problems involving fractures and thermal shocks.