## 2 Materials and Methods
### Interaction Laws
This is due to the microdestruction of the soil structure in the contact layer in the case of undisturbed soil and the rearrangement of soil particles in the case of disturbed soil. In the case of consolidated soil, with an increase in the value of relative displacement, the value of \(\tau\) rose to \(\tau_{p}\), reached at \(u\ =\ u_{p}\). In the section \(0\leq u\leq u_{p}\), the soil structure underwent microdestruction. After reaching \(\tau_{p}\), the value of shear stress dropped rapidly to the value of \(\tau_{r}\) at \(u\ =\ u_{*}\) and remained constant. In the section \(u_{p}\leq u\leq u_{*}\) of curve \(\tau(u)\), an intensive destruction of the soil contact layer occurred. Accordingly, the values of shear stress dropped rapidly. After complete destruction of soil structure, the value of \(\tau\) remained constant and equal to \(\tau\ =\ \tau_{r}\). In the section \(u>u_{*}\) the [PERSON]'s law of dry friction was obviously fulfilled. Here the value of \(\tau\) was determined from the relationship  
\[\tau\ =\ c+f\sigma_{N} \tag{7}\]  
where \(c\) is the cohesion force between soil particles, \(f\) is the coefficient of friction between soil particles, and \(\sigma_{N}\) is the stress normal to the contact surface, as was noted above. Note that in Equation (5), the friction coefficient \(f\) is taken precisely between the soil particles and not between the soil particles and the pipeline's outer surface. This is true in cases where soil particles have adhered to the pipeline's outer surface, and strong bonds have formed between them. In cases where this has not yet happened, i.e., when soil particles have not had time to adhere to the pipe surface and the soil has not yet consolidated after the pipeline construction, the interaction process \(\tau(u)\) occurs along the curves 1, 2, and 3. In this case, the peak value of shear stress \(\tau_{p}\) did not appear and as the value of relative displacement \(u\) increased, the value of \(\tau\) increased to \(\tau\ =\ \tau_{r}\) and remained constant. Note that after the destruction of the soil contact layer, the repeated processes of the pipeline interaction with soil occurred along the curves 1, 2, or 3. Longitudinal interaction of the underground pipeline with soil occurred in two directions conditionally, in direct motions and reverse motions. On the diagram \(\tau(u)\), the reverse motion of the pipeline relative to soil can begin from any point of it, depending on the change in seismic load. The reverse and repeated direct motion of the pipeline relative to soil occurs in the most complicated way depending on the degree of the contact layer destruction [14]. This follows the results of experimental studies of the interaction of underground pipelines with soil and studies of the aftermath of strong earthquakes on underground utilities [14]. In addition, experimental studies have indicated that the value of the peak shear stress (the force of interaction) \(\tau_{p}\) depends significantly on the interaction velocity \(\frac{du}{dt}\). Depending on the value of the interaction velocity and the strength of structural bonds between soil particles, the value of \(\tau_{p}\) can be almost twice as large as the Coulomb value \(\tau_{r}\). Experiments show that the values of \(u_{*}\) do not depend on the interaction velocity [14]. In the diagram \(\tau(u)\) (Figure 1), in the section \(u>u_{*}\) where the [PERSON]'s law (7) is fulfilled, the values of the interaction force \(\tau\) uniquely depend on the normal stress \(\sigma_{N}\). This section is the limiting stage of the pipeline interaction with soil. It follows the prelimiting stage in section \(0\leq u\leq u_{*}\). These stages of interaction are two stages of the same interaction process.