## 3 Results and Discussion
#### 3.1.1 The Case with the Operating Pressure P\({}_{1}\) = 7.5 MPa
Figures 14-16 show the displacements along the pipeline at the time periods \(t=1\) ms, \(t=2.25\) ms, and \(t=3.5\) ms respectively. The analysis of the results showed that, within the time \(t=1.0\) ms, the size of the crack maintained its length of 100 mm. At the same time, the wall spacing increased and reached the value of 23 mm at the free edge. Further, within the time \(t=2.25\) ms, the wall spacing at the free edge increased by a factor equal to 2, while it increased  
Figure 6:0 Displacements at time \(t=3.5\) ms. Figures 17-19 show the plastic strains along the pipeline at the time periods \(t=1.0\) ms, \(t=2.25\) ms, and \(t=3.5\) ms respectively. The results show that, up to the time \(t=1.0\) ms, any plastic deformation did not develop in the crack tip area. This was consistent with the fact that no crack elongation was observed before this time. Within the time \(t=2.25\) ms, plastic strains were directly concentrated in the crack tip area. The growth of the crack occurred along the pipeline. Moreover, within the time \(t=3.5\) ms, the plastic strains were concentrated along the lateral surface of the crack; thus, their growth stopped. The growth of the crack at its tip was noticed from the time \(t=1.0\) ms up to that equal to \(t=3.5\) ms, and from 100 mm until 580 mm in length. This was 3.4% higher than in the numerical calculation without taking the pressure on the crack-side walls into account. Figure 6:1 Plastic strains at time \(t=1.0\) ms.  
Figure 6:2 Plastic strains at time \(t=3.5\) ms.  
Figure 6:3 Plastic strains at time \(t=2.25\) ms. The aforementioned work presented one part of the authors' research. On the basis of finite-element simulations performed by ANSYS-19.2 [68], the study of the fracture of steel gas pipelines under the influence of internal pressure on the propagation of an incipient crack in their tip area was implemented on the existing gas pipeline \"Beineu-Bozoy-Shymkent\" [59]. A cylindrical shell finite-element model was initially created (Figures 1, 3 and 4), and a step-by-step modeling technique was developed by the software package (Figure 2). Furthermore, the mechanical parameters of the steel material of the full-scale gas pipeline were determined [59], in which a qualitative finite-element model was similarly created (Figures 1, 3 and 4), and a stepwise methodology of modeling within the software package was developed (Figure 2). Besides, the mechanical characteristics of the material of the natural gas were achieved (Section 2.1). Regarding the propagation of the incipient crack, it was established that, with an internal working pressure of 7.5 MPa (Section 3.1.1) at the beginning of the crack growth with a time indicator \(t=1\) ms, the crack expanded (displacement) in the circumferential direction by 0.036 m (Figure 6). From the time \(t=2.25\) ms up to \(t=3.5\) ms, the crack started to grow up along the longitudinal direction, while the crack growth due to the decompression of the structure stopped within the time \(t=3.75\) ms (Figures 9 and 11). According to these findings, the crack on the gas pipeline increased by 5.6 times, i.e., from 0.2 m up to 1.12 m in the longitudinal direction. The value of the equivalent stress at time \(t=1.0\) ms was equal to 295 MPa (Figure 7). Considering the passage of time from \(t=2.25\) ms up to \(t=3.5\) ms, the equivalent stresses increased up to 453 MPa (Figures 8 and 10). Besides, at the time \(t=3.75\) ms, the growth of the equivalent stresses stopped due to the gas decompression. Consequently, within the time \(t=3.5\) ms, the equivalent stresses increased by 1.53 times. The results corresponding to the maximum pressure of 9.8 MPa (Section 3.1.2) showed that the increment of the internal pressure led to an increase in stresses in the pipeline and, accordingly, a decrease in the time of crack development.