Simulation of soil behavior under longitudinal motion of underground pipeline in one-dimensional statement

. The process of shear interaction of an underground pipeline with soil is numerically modeled in a one-dimensional statement. The main attention is paid to the behavior of soil around the pipeline. The deformation characteristics of the pipeline are ignored. A technique for the numerical solution of a one-dimensional problem of soil behavior and shear wave propagation using the finite difference method was developed. The developed method of numerical research made it possible to take into account the nonlinear properties of soil strain and it was tested for elastic and viscoelastic simulation of soils. Numerical results were obtained for the shear wave propagation in elastic and viscoelastic soil under the longitudinal motion of an underground pipeline. The results obtained showed the attenuation of the wave parameters with distance from the cross section of the underground pipeline. The attenuation of waves with distance is justified by the dissipation of the deformation energy on an expanding cylindrical soil layer. In the case of viscoelastic soil, attenuation in time in fixed points of soil is also observed, the maximum shear stresses are reached near the contact surface of the underground pipeline, and in this area, an intensive decrease in the shear stress amplitude is observed.


Introduction
Under shear interaction of underground pipelines with soil, shear stresses arise at the contact boundary and these stresses propagate in soil. Depending on the nature of the propagating waves, it is possible to determine the soil conditions around the underground pipeline, potential accidents in pipelines, and soil response to other objects in soil [1][2][3][4][5]. Therefore, the study of the problem of wave propagation in soil under interaction with underground pipelines can be considered an urgent task. The strength and seismic resistance of underground pipelines also largely depend on the stress-strain state of soil and, naturally, on the interaction force that occurs when the pipeline moves relative to soil [4][5][6][7][8][9][10][11]. Therefore, defining the stress state of soil around the underground pipeline is also necessary to assess the strength of the underground pipeline.
The propagation of shear waves in various types of soil was experimentally and theoretically studied in [9][10][11][12][13][14][15][16]. In articles [4][5][16][17], the method of characteristics and finite difference method for elastic and elastic-plastic media were used. The shear interaction of underground pipelines with soil and the wave propagation in soils were studied in [9][10][11]16] in a two-dimensional formulation. From the solution of these problems, it was determined that the stress and strain, and the amplitude of velocities arising behind the wave front when the wave propagates in elastic media, retain the wave profile and no wave attenuation is observed. The absorption of longitudinal waves occurs when the viscous properties of the medium are taken into account [4,[18][19]. If the viscous properties are considered, the values of the wave amplitude decrease with respect to distance and time, and the maximum values of stress and strain are reached at different times due to the rheological properties of strain. However, the results obtained in [9,16] for two-dimensional problems are difficult to analyze the soil behavior, sometimes it becomes impossible. In the studies of cylindrical shear wave propagation in elastically modeled soils [4,[17][18], when an impact load is set, the characteristic lines are straight, the front is a shock front, and the wave parameters have a discontinuity. Therefore, when such conditions are set at the boundary, the solution of problems by the method of characteristics is the most appropriate one. If an arbitrary load is specified at the boundaries or if it is required to consider the nonlinear properties of soil, then it is much more difficult to build solutions on the characteristic lines. In the case of considering the nonlinear properties of soil, it is difficult to determine the characteristic lines and build solutions on the characteristic domain; sometimes it is impossible to obtain analytical solutions even considering structural changes or soil damage.
This article is devoted to the study of the one-dimensional cylindrical wave propagation in soils during the movement of an absolutely rigid cylindrical underground pipeline in the direction of the symmetry axis. This is a continuation of the studies presented in [15], where the propagation of angular shear waves originated under the rotational motion of an underground pipeline was considered. The aim of this study is to develop a solution technique that allows applying arbitrary loads at the boundary and complex strain properties, and determining the parameters of cylindrical shear wave propagation in elastic and viscoelastic soil.

Problem statement
Let us assume that there is an extended and rigidly fixed underground pipeline with outer radius 0 r r = in the unbounded soil. Let at the initial time the underground pipeline begin translational motion in the direction of the axis of symmetry, while the deformation of the pipeline is ignored, i.e. we consider it an absolutely undeformable body. In this case, shear cylindrical waves begin to propagate in soil, the parameters of these waves are axisymmetric with respect to the axis of the pipeline, and they depend only on the radial coordinate and time, i.e. the problem is one-dimensional. Therefore, we consider a onedimensional problem in a cylindrical coordinate system. The equation of ground motion in the absence of body forces in the Euler representation has the following form: where r is the radial coordinate, is the velocity of soil particles along the z cylindrical coordinate, and To obtain a closed system of equations, it is necessary to add the equation of state into (1)- (2). We write the equation of state in the general form where is the shear strain rate. Thus, the closed system of equations consists of equations (1) To solve this system of equations, we accept the following initial conditions: (4) and boundary conditions: where ( ) are the displacement and velocity of the underground pipeline in the direction of the extent axis ( Oz ).
We solve the problem posed by the finite difference method using the schemes from [15,20]. The choice of a numerical method for solving the problem is due to the fact that in the future, when solving the problem of the interaction of an underground pipeline, it would be possible to take into account complex strain properties in the equation of state of soil, such as structural destruction, water saturation, viscoplasticity, and others [15].

Method of problem solution
Let us use the finite-difference scheme [20] and compose finite-difference relations for the problem posed. Unbounded soil is considered to be limited by radius R r = . Then, considering the velocity of shear waves propagation S c , the solution to the problem under consideration is obtained before the front of the leading wave reaches the boundary R r = , i.e. we obtain a numerical solution up to the time point . We divide the radial segment 0 r R − into N infinitely small parts, i.e. cylindrical cells. We introduce the following notation. We denote the values of particle velocities at the nodal points of the cells 0 r r r k  = at the points in time . The values of shear stresses and shear strains determined at the centers of the cells We assume that up to a certain point in time n t t = , the values of all parameters of the problem are known. We find the same parameters at the next time steps. Using the finite difference scheme [20], from equation (1), we determine the particle velocities for the time point at the nodal points of the cylindrical cell as follows: where n t  is the time step; Shear strain is calculated using the finite difference relation: where the strain rate from (2) From formulas (8) and (9), we know the values of the shear strain and its velocity, determined at the center of the cylindrical cell at time points Here, when determining shear stresses (10) 2. The time step, satisfying the stability conditions of the difference scheme, is chosen in the following form [20]: During the calculation, it is possible to increase the time step. At that, we limit the step increase to no more than 10%, that is, if , then we take . Once the step 2 / 3 +  n t is determined by formula (11), we can determine the time step as follows: So, meeting the stability conditions (11) and using the values of velocity, strain and stress at previous times, we determined the values of these parameters for the next time point (6)- (10). By performing these actions in a sequential manner, it is possible to determine the parameters of the shear wave propagating in the surrounding soil up to time . This is the main point of the numerical solution of the problem posed. If an increase in the wave propagation time is required, then we must increase the size of the considered domain of the sought-for solutions, i.e. R r = .

Numerical results and their analysis
According to the algorithm for the numerical solution of the problem, a program was compiled. Using this program, solutions were obtained with the following initial data: initial density of soil -

Elastic Shear Waves
If the soil is modeled by an elastic law of deformation, then the specific form of this law under shear is ) ( are the reference values of shear stress and shear strain before the deformation. In this case, relation (10) has the following form:  As seen from Fig. 1, in the case of setting the tangential velocity at the boundary 0 r r = , the amplitude of particle velocity and shear stress decreases with distance. Amplitude attenuation occurs at the first arrival of the wave; then, in fixed cross sections, no decrease in time is observed. With a positive value of the particle velocity, the value of shear stress increases in absolute value. In the case of reverse movement, i.e. at a negative value of the particle velocity, the tangential stress decreases and the stress state is unloaded.
Note that a similar decrease in particle velocity and shear stress was observed during rotational motion of an underground pipeline in [15]. Thus, under the elastic strain of soil, the shear wave attenuates with distance. Energy dissipation occurs due to its redistribution on an expanding cylindrical layer, as was said in [15].

Viscoelastic Shear Waves
Various viscoelastic deformation models can be used when considering the viscous characteristics of soil. In the case of using the Kelvin-Voigt model:   Figure 3 shows the shear stress-shear strain diagram, obtained by solving problems in the same fixed points of soil as in Fig. 2. As seen from Fig. 3, the shear stress-shear strain diagram, in this case, describes the nonlinear deformation of soil around the underground pipeline and the "hysteresis loops" (the area bounded by the loading and unloading curves), which indicate the energy dissipation for each deformation cycle. The greatest loss of deformation energy occurs in soil near the underground pipeline. Figure 4 shows the distribution of maximum values in modulus of shear stress with distance. It shows the intensity of the decrease in the amplitude of shear stress depending on the radius, and the maximum shear stresses achieved near the contact surface of the underground pipeline. A decrease in velocity frequency of an underground pipeline has practically no effect on the velocity frequency of soil particles, and slightly reduces the intensity of the amplitude attenuation with distance ( Fig. 5(a)). In this case, the amplitudes of the shear stress values increase ( Fig. 5(b)) in comparison with the dependence shown in Fig. 2(b). An increase in the period of pipeline motion also leads to an increase in energy loss for each deformation cycle (Fig. 6). Figure 7 shows the distribution of the maximum values in terms of shear stress modulus, reaching an extremes at the considered time interval, with a distance from the pipeline cross section. Here it can also be seen that the maximum shear stresses are reached near the contact surface of the underground pipeline, and in this area, there is an intensive decrease in the amplitude of shear stress.

Conclusion
A one-dimensional problem on the behavior of soils and the shear waves propagation in the process of shear interaction of an underground pipeline with soil was posed. A numerical technique for solving the problem posed using the method of finite differences with a central difference scheme was developed. The developed method of numerical research makes it possible to consider the nonlinear properties of soil strain; it was tested for elastic and viscoelastic modeling of soils. Numerical results of the shear wave propagation in elastic and viscoelastic soils were obtained. The compiled algorithm was implemented on a PC using Java. The results are presented in the form of graphs. The results obtained showed the attenuation of the wave parameters with distance from the cross section of the underground pipeline. The attenuation of waves with distance is justified by the deformation energy dissipation on an expanding cylindrical soil layer. In the case of viscoelastic soil, attenuation in time was also observed in fixed points of soil, the maximum shear stresses were reached near the contact surface of the underground pipeline, and, in this area, there was an intense decrease in the amplitude of shear stress. The energy dissipation for each cycle of deformation for a viscoelastic soil was shown.