## 2 Materials and Methods
### Interaction Laws
They also indirectly confirm the existence of the soil contact layer and the above hypothesis. It was shown in [38] that when the equation of state (the law of deformation) of soil takes into account the process of soil destruction under strain [39; 40], the classical boundary condition of complete cohesion (\(u~{}=~{}0\)) is applicable at the direct contact boundary of soil medium with a rigid body. In this formulation, the forces of interaction are formed in the process of soil strain around the pipeline in accordance with the accepted equation of the state of soil. The law of soil deformation should be adequate to the process of deformation and destruction of the soil contact layer [35; 38]. However, in this case, it is necessary to jointly consider the deformation process in the soil environment and underground pipeline. This leads to complex coupled wave problems for two media. Under the action of seismic waves, even for the simplest one-dimensional wave problems, the theoretical problem's solution is quite laborious and can be obtained using numerical methods only [35; 38]. This circumstance requires a search for simpler methods to determine the stresses in an underground pipeline under seismic effects. However, the growing economic and environmental requirements for underground pipelines as the transportation means for liquid and gas substances (sometimes highly toxic ones) [41] pose a problem for researchers to reliably and accurately determine and predict the stresses occurring under strong ground motion. Experimental determination of seismic stresses in underground pipelines is costly and very laborious. Therefore, theoretical methods for determining stresses under seismic impacts in underground pipelines are more appropriate. However, the option shown above requires, first of all, more accurate and reliable knowledge of the law of interaction of an underground pipeline with soil. The more reliable this law is, the more accurate the definition of the value stresses in underground pipelines under seismic effects. Consider here the law of longitudinal interaction of the pipeline with soil. According to the results of experiments and observations [1; 2; 3; 6; 7; 14; 37], schematic changes in dependencies \(\tau(u)\) are shown in Figure 1.  
Figure 1: Schematic dependencies of the shear stress \(\tau(u)\) arising on the contact surface of the pipeline with the soil on relative displacement \(u\) at various normal stresses. The experiments on the longitudinal interaction of pipelines with soil and the shear interaction of rigid bodies with soil, presented in [14], show that the process of interaction of a pipeline with soil of disturbed and undisturbed structure differs significantly. Structurally disturbed soil during the construction of pipeline communications and facilities consolidates over time and forms a new coherent soil structure around the pipeline. When the pipeline interacts with this soil layer, the interaction process occurs along complex curves 1-3 (Figure 1). Curves 1-3 refer to different values of the stress \(\sigma_{N}\) normal to the outer surface. Curves 1-3 refer to stresses \(\sigma_{N1}<\sigma_{N2}<\sigma_{N3}\), respectively, and are discussed below. As seen from Figure 1, the dependences \(\tau(u)\) in the range of relative displacements \(0\leq u\leq u_{*}\) for disturbed (curves 1-3) and undisturbed soil structures (curves 1-3) differed qualitatively and quantitatively. As noted above, in the pipeline interaction with soil, a soil contact layer was formed, and the structural destruction of soil occurred precisely in this contact layer. Under interaction with soil, when the contact layer was structurally undisturbed, the dependences \(\tau(u)\) show the peak value of the interaction force, and when the contact layer of soil was structurally disturbed in this area, the peak value of \(\tau_{p}\) did not appear (curves 1-3). In the section \(0\leq u\leq u_{*}\), the change in \(\tau(u)\) in both cases was curvilinear.