## 2 Materials and Methods
### Interaction Laws
In this case, the sign of the friction force on the outer surface of the underground pipeline in contact with soil is determined with account for the interaction velocity \(\frac{du}{dt}\) according to the law  
\[\tau\ =\ \chi K_{x}(\sigma_{N})u \tag{11}\]  
where \(\chi\ =\ sgn(v)\), is the rate of relative displacement, determined by the relationship  
\[v\ =\ v_{g}-v_{c}\ =\ \frac{du_{g}}{dt}-\frac{du_{c}}{dt} \tag{12}\]  
Then  
\[ \begin{array}{l}\chi\ =\ +1\ at\ v>0\\ \chi\ =\ -1\ at\ v<0\\ \chi\ =\ 0\ at\ v\ =\ 0\end{array}  \tag{13}\]  
According to (13), when the soil deformation around the pipeline is greater than the deformation of the underground pipeline itself (\(\varepsilon_{g}>\varepsilon_{c}\)), the interaction force \(\tau\) is an active force involved in motion. At \(\varepsilon_{g}<\varepsilon_{c}\), the force of interaction is a passive one resisting to friction force. This circumstance is of key importance when solving the problems of seismic resistance of underground pipelines. Depending on the property of the interaction force (active or passive one), the stress state of the underground pipeline is formed in different ways. The mechanisms of formation of longitudinal stresses in an underground pipeline were considered in [35]. Dynamic theory does not provide a priori the possibility of converting the interaction force from active to passive one and vice versa. As seen from Figure 1, this is possible only with the values of relative displacements \(u<u_{p}\). As noted above, in this case, the relative displacements are insignificant. Considering that during strong earthquakes, the absolute displacement of soil reaches 0.3-0.4 m, it becomes obvious that the dynamic theory is inapplicable to these cases without taking into account relations (11)-(13). In [22; 23; 24], the cases were considered when the soil surrounding the underground pipeline has greater rigidity than the pipeline material (\(c_{g}>c_{c}\), where \(c_{g}\) is the propagation velocity of longitudinal waves in soil, \(c_{c}\) is the propagation velocity of longitudinal waves in the pipeline). This hypothetical assumption also does not hold in practice. Some interesting results could be obtained under such assumptions when solving the problems of seismic resistance of underground pipelines [22; 23; 24]. Since the soil environment around the pipeline is always less rigid than the underground pipeline, the soil contact layer always deforms more than the underground pipeline under seismic waves propagation in the underground pipeline-soil system. Consequently, the interaction force acting on the pipeline has more pronounced active properties than passive ones [16]. When a monochromatic seismic wave propagates along an elastic or viscoelastic pipeline (not considering the soil effect) at a stress amplitude of \(\sigma_{max}\ =\ 0.5\) MPa, no stresses greater than 0.5 MPa appeared in the pipeline. The formation of large longitudinal stresses (100 MPa or more) in underground pipelines, leading to the pipeline destruction [4], is exclusively the result of significant deformations of soil surrounding the underground pipeline. As shown above, the deformation properties of soil surrounding the underground pipeline, or its stress-strain state under strong ground motion, is of significant determining importance in the formation of the stress-strain state of an underground pipeline. The nonlinearity of the diagram \(\tau(u)\) (Figure 1) and the manifestation of the peak value of \(\tau_{p}\) in the diagram occur as a result of the soil contact layer destruction [38; 42]. The soil with weak structural bonds, under shear deformations, begins to undergo microdestruction, which leads to a nonlinear change in the diagram \(\tau(u)\), already in the section \(0\leq u\leq u_{p}\)In the section \(u_{p}<u\leq u_{s}\), the structure of the soil contact layer is completely destroyed.