## 2 Materials and Methods  
Calculation Model of the Stress-Strain State and Fracture of a Steel Gas Pipeline Section with an Incipient Crack-like Defect  
A section of an aboveground main steel gas pipeline with an incipient crack-like defect between two supports was considered (Figure 1). Support \(A\) implemented the conditions of rigid pipeline embedment, while support \(B\) was a longitudinal-moving support. It was also assumed that the distance between the supports was \(l_{\text{TP}}=36.0\) m. The incipient crack-like defect was located in the central part of section No. 2. It is directed along the pipe perpendicular to the axial section. The crack-like defect has the shape of a cutout on the outer pipe surface with a length of 200 mm, a width of 1 mm, and a depth equal to 8 mm. In this case, the supports did not influence the process of crack development. The geometrical parameters of the pipeline were those of the natural gas pipeline \"Beineu-Bozoy-Shymkent\" in the Republic of Kazakhstan, including its underground and aboveground parts [59]. The pipe diameter was \(d_{\text{TP}}=1047\) mm, while the thickness of the pipe wall was \(h_{\text{TP}}=15.9\) mm. The gas pipeline was made of X70 steel and produced based on the elastoplastic material's behavior. The physical and mechanical characteristics of the steel material were used according to the standard [66]: density \(\rho=7810\) kg/m\({}^{3}\); elastic modulus \(E=2.06\) GPa; yield strength \(\sigma_{\text{T}}=505\) MPa; and ultimate strength \(\sigma_{\text{B}}=570\) MPa. Research was carried out for a temperature of +22 \({}^{\circ}\)C. According to these data, the stress-strain state of the gas pipeline was investigated by considering the elastic-plastic properties of the steel material. The state equation for the elastic state of the material takes the form \(\sigma_{eq}=E\cdot\varepsilon_{eq}\) for \(0\leq\sigma_{\text{d}}\leq\sigma_{\text{T}}\), where \(\sigma_{eq}\) and \(\varepsilon_{eq}\) are von Mises equivalent stress and strains respectively. If \(\sigma_{\text{T}}\leq\sigma_{eq}\leq\sigma_{\text{B}}\), the model plastic behavior of the steel is taken into consideration. The plastic behavior of the steel was particularly described by the Bilinear Isotropic Hardening (BIH) model [67]. The state equation for the BIH-plastic state of the material instead takes the form \(\sigma_{eq}=\sigma_{\text{T}}+H(\varepsilon_{eq}-\sigma_{\text{T}}/E)\), where \(H=d\sigma_{eq}/d\varepsilon_{eq}\) is the hardening modulus. When the strain rate exceeded the values of \(10^{-2}\) s\({}^{-1}\), the plastic behavior of the steel, described by the Cowper Symonds Strengh (CSS) model, was assumed [67]. The use of this model allowed us to consider the dependence of the mechanical characteristics of the X70 steel on the strain rate. The state equation for the CSS-plastic state of the material takes the form: \(\sigma_{eq}=\left[A+B\left(e_{eq}^{pl}\right)^{n}\right]\cdot\left[1+\left(D^ {-1}\cdot\partial e_{eq}^{pl}/\partial t\right)^{1/q}\right]\), where \(A\) is the yield stress at zero plastic strain; \(B\) is the strain hardening coefficient; \(e_{eq}^{pl}\) is the plastic strain; \(n\) is the  
Figure 1: Steel gas pipeline model: (**a**) a general view and (**b**) a diagram of the pipe section with a crack-like defect between the supports. strain hardening exponent; \(\partial e_{eq}^{\textit{I}}/\partial t\) is the plastic strain rate; whilst \(D\), \(q\) are the strain rate hardening coefficients [66; 67]. Material failure was modeled based on the Maximum Stress Criterion (von Mises Criterion). When equivalent stresses are higher than the ultimate stress \(\sigma_{eq}>\sigma_{\text{T}}\), local destruction of the pipeline material occurs. The operating pressure \(P_{1}\) is 7.5 MPa, while the maximum pressure \(P_{2}\) is 9.8 MPa. The main issue was solved using dynamic formulations.