## 3 Results and Discussion
#### 3.1.1 The Case with the Operating Pressure P\({}_{1}\) = 7.5 MPa  
The finite-element modeling of the strain of the thin-walled pipeline section without a crack at working pressure showed that the pipeline structure deformed in the elastic area. Figure 6 shows the displacements at the beginning of the growth of the longitudinal crack, i.e., at the time \(t\) = 1 ms. Within this time, the crack circumferentially expanded under the action of the internal pressure. At the same time, the crack deepened to the full thickness of the pipeline. Besides, from the time \(t\) = 1 ms, plastic strains began to develop at the crack tips, and the crack started to grow along the pipeline. Figure 7 shows the equivalent stresses at time \(t\) = 1.0 ms. The variation of the stress state in the crack area was locally characterized. Considering the passage of time, the crack grew up along the longitudinal direction. Within the time \(t\) = 2.25 ms, the stresses at the crack tips reached their maximum value (Figure 8). Figure 5: Schematic diagram of the gas pipeline (Figure 1) for the finite-element simulation with a length of 7.0 m.  
Figure 6: Displacements at the moment of initiation of the longitudinal crack-like defect growth. Figure 7: Equivalent stresses at the moment of initiation of the longitudinal crack-like defect growth. Plastic strains developed at the crack tips. The remaining part of the pipeline deformed in the elastic area. Figure 9 instead shows the plastic strains at time \(t=2.25\) ms. The localization of the plastic strains in the crack tip areas was clearly observed. With the passage of time, the qualitative picture of the localization of the plastic strains in the crack tip area did not change. The maximum stresses at the crack tips were held from the time \(t=2.25\) ms up to the time \(t=3.5\) ms. During all of this period, the crack grew up along the longitudinal direction. Figure 10 shows the equivalent stresses at time \(t=3.5\) ms. Within the time \(t=3.75\) ms, the growth of the crack along the longitudinal direction stopped. Figure 11 shows the displacements at the final period of the crack growing, i.e., at the time \(t=3.75\) ms. According to these numerical results, within the time \(t=3.75\) ms, the crack grew up along the pipeline from \(0.2\) up to \(1.12\) m.  
Figure 8: Equivalent stresses at the time \(t=2.25\) ms.  
Figure 10: Equivalent stresses at time \(t=3.5\) ms.  
Figure 9: Plastic strains at the time \(t=2.25\) ms.  
3.1.2. The Case with the Maximum Pressure P\({}_{2}\) = 9.8 MPa  
The increment in internal pressure brought an increase in stresses along the pipeline and, consequently, a decrease in the time of crack development. Figure 12 shows the displacements at \(t\) = 0.75 ms, i.e., at the time of the initiation of the growth of the longitudinal crack. The stop of the growth of the longitudinal crack was conversely observed at time \(t\) = 3.5 ms (Figure 13). Together, the process of the crack growing along the longitudinal direction at the maximum pressure qualitatively slightly differed from the process of the crack growing at the working pressure. According to the finite-element results, the crack grew from 0.2 up to 1.18 m within the time \(t\) = 3.5 ms.  
Figure 11: Displacements at the moment of stopping the growth of the longitudinal crack-like defect. Figure 12: Displacements at the moment of initiation of the growth of the longitudinal crack-like defect at the maximum pressure. Figure 13: Displacements at the moment of stopping the growth of the longitudinal crack-like defect at the maximum pressure. Results on the Stress-Strain State and Destruction of the Gas Pipeline in the Area of the Tip of the Crack-like Defect  
The development of a through-crack in the area of its tip under the action of dynamic loading was considered [44]. The dynamic pressure, equal to \(P_{1\text{max}}=7.5\) MPa, was applied as shown in Figure 4. An open crack with a length of 100 mm and a wall spacing of 1.0 mm was considered.