## 5 Numerical Results and Discussion
### Simulation and Experimental Verification
#### 5.2.2 Stress and Strain Field near the Crack in the Welding Heating and Cooling Processes  
The geometry and boundary conditions of the heating process are also shown in Figure 13. The crack is formed in the welding seam zone at a time of 12 s in the welding process. The isothermal crack is predefined with the length of 8 mm (the crack area is between \(x=16\) to \(x=24\) with \(y=0\)). The stress field near the crack tip is observed as beginning from a welding heating time of 12 s. Figure 17 is the element distribution of the stress field near the crack tip for observation. Eight elements near the crack tip stress are selected to be drawn as a curve. Figure 18 is the time history curve of the stress of the eight elements that began from a welding heating time of 12 s. The values and general trends of stress at different times can be mastered by observation of Figure 18.  
Figure 16: A plate with an inclined isothermal crack at the cooling process: (**a**) the distribution of heat flux; (**b**) the distribution of temperature; (**c**) the distribution of stress \(\sigma_{x}\); (**d**) the distribution of stress \(\sigma_{y}\); (**e**) the distribution of stress \(\sigma_{xy}\).  
Figure 17: Element distribution of the stress field near the crack tip for observation (The colors represent the intensity of the stress field, the red represents the largest, the blue represents the smallest).  
It can be seen from Figure 18 that the stresses of elements 1, 2, 3 and 6 are all linear in growth and obvious stress concentration. The reason is that the crack tip is affected by the thermal expansion and hampered by the nearby area of metal. The crack tip affected by the temperature becomes weak as the welding heat source moves far away from the crack. The stress values fall back at the end of welding. The stress values of elements 3, 4, 2 and 8 fluctuate obviously compared to that of elements 1, 2, 6 and 6. The maximum principal stress values of elements 4, 5, 2 and 8 fluctuate  
Figure 18: The stress curves of the elements near the crack tip in the heating process. (**a**) The stress curve of element 1; (**b**) The stress curve of element 2; (**c**) The stress curve of element 3; (**d**) The stress curve of element 4; (**e**) The stress curve of element 5; (**f**) The stress curve of element 6; (**g**) The stress curve of element 7; (**h**) The stress curve of element 8.  
significantly. The reason is that the heat flux distribution is interfered by the discontinuity of the crack, which causes the fluctuations of the temperature and the stress value.  
Figure 19 is the time history curves of stress of the eight elements in the welding cooling process. The changing conditions of stress of the eight elements in the welding cooling process can be observed in Figure 19.  
Figure 19: The stress curves of the elements near the crack tip in the cooling process. (**a**) The stress curve of element one (**b**) The stress curve of element two; (**c**) The stress curve of element three; (**d**) The stress curve of element four; (**e**) The stress curve of element five; (**f**) The stress curve of element six; (**g**) The stress curve of element seven; (**h**) The stress curve of element eight.  
It can be seen from Figure 19 that the stress of the elements near the crack tip all decrease rapidly as the cooling progresses. The reason is that the strong transient cooling effect is produced when the welding heat source suddenly disappears at the beginning of cooling. The cooling rate of the welding seam zone is the fastest. The stress value of the welding seam zone is changed rapidly. The decreasing rate of stress becomes slow after cooling in a period of time. The cooling rate becomes lower as the temperature difference becomes small. The welding stress value decreases slowly as the temperature becomes low. The final stress value is not zero when the cooling is over, at which point the final residual stress exists.