## 2 Materials and Methods  
### Interaction Laws  
Consider the laws of longitudinal interaction. The seismic load acting on an underground pipeline along an extended pipeline is considered known in the dynamic theory, taken as a function \(u_{g}(x,t)\), and enters Equation (4). Since the length of a longitudinal seismic wave can be significant (from hundreds of meters to a kilometer), in most cases, soil displacement is considered depending on time \(t\) only (not on spatial coordinate \(x\)), in the form of a function [33; 34]  
\[u_{g}\ =\ u_{m}sin(\omega t) \tag{5}\]  
or  
\[u_{g}\ =\ u_{m}\cos(\omega t) \tag{6}\]  
where \(u_{m}\) is the amplitude of longitudinal soil displacement and \(\omega\) is the oscillation frequency of seismic wave. The value of \(u_{m}\) is determined from seismic data of instrumental records of ground motion during various earthquakes. Depending on the earthquake intensity and soil type, its value reaches 0.1-0.4 m. When \(u_{g}(x,t)\) is known, the displacement of the pipeline sections \(u_{c}\) is determined from one-dimensional equations of the pipeline vibration; then, longitudinal strains and stresses are determined along the pipe. In this statement of the problem, when determining the longitudinal seismic stresses in an underground pipeline under longitudinal seismic loads, the stress value depends on two parameters: soil displacement and mechanical properties of soil. The latter parameter is taken into account through the coefficient \(K_{x}\) in (4). The value of \(K_{x}\) is determined experimentally from the results of corresponding experiments [14]. Usually \(K_{x}\) is called the coefficient of interaction [19; 20; 35; 36]. According to (4), the coefficient \(K_{x}\) has the dimension MPa/m and physically means the rigid bond between soil particles and the pipeline's outer surface in contact with soil. This rigid bond takes place in the longitudinal, transverse, and other directions. It should be noted that there is a strict application area for the law or model (4) [14]. Equation (4) is valid up to the value of relative displacement \(u\ =\ u_{*}\), at which a soil particle is separated from the outer surface of the pipeline, the bond between the soil particles and the outer contact surface of the pipeline ceases to exist. Further, a different law is needed to describe the process of a pipeline interaction with soil. As shown, for example, in [14; 37], the process of interaction beyond of \(u_{*}\) is described by the Coulomb law. In fact, due to cohesion processes between soil and the outer surface of the pipeline [1], the interaction occurs between the soil particles \"stuck\" to the pipeline and other particles of soil. Therefore, under longitudinal interaction of the pipeline with soil, there is, in fact, a soil shear and cut-off near the outer surface of the pipeline in a certain soil layer. It is difficult to state the existence of such a soil layer in experiments [14; 38]. In [38], the existence of a soil contact layer under shear interaction of a rigid body with soil was substantiated theoretically. It was shown that at insignificant stresses \(\sigma_{N}\) normal to the body's outer surface, the contact layer's thickness was about 0.01 m.  
The process of interaction force \(\tau\) formation in a given contact layer was shown in [14; 38], first in proportion to the value of relative displacement \(u\), and then after reaching equality \(u\ =\ u_{*}\) according to the Coulomb law. It was shown there that the process of shear strain of the soil contact layer occurs separately. Under shear interaction of a rigid body with soil, the contact layer can be destroyed, while the soil outside the contact layer remains intact. This means that the strain in the contact layer does not obey elastic laws, and the soil medium, as a whole, can remain elastic or viscoelastic during earthquakes. The results of a survey of strong earthquake aftermath in underground pipelines show the existence of a solid, cohesively stuck thin layer of soil on the outer surface of the pipeline [2; 3].