Longitudinal wave propagation in an extended cylindrical body with external Coulomb friction

. The propagation of elastic longitudinal waves in an extended cylindrical body located inside an unstrained body and interacting according to the Coulomb law is considered in the article. The problem is studied in a two-dimensional statement; therefore, the friction force (i.e., the interaction conditions) is included in the system of equations as a boundary condition. The Coulomb friction force arises due to the deformation of a cylindrical body. The reliability of numerical calculations is substantiated by solving test cases and comparing the calculations with experimental results. The numerical results obtained are presented in the form of graphs and analyzed. It is shown that the parameters (stresses and strains) of waves propagating in an elastic cylindrical body with external Coulomb dry friction decay with distance. The mechanism for reducing the stress-strain state and wave parameters is explained by the consumption of elastic energy to overcome the friction force that occurs on the contact surface. The results of the two-dimensional problem are also compared with the results of a similar problem in the one-dimensional theory, where the friction force enters directly into the equations of motion. The deviations of the results of the one-dimensional theory are up to 8-15% depending on the accepted values of the friction coefficient, i.e. the violation of the plane section hypothesis taken in one-dimensional calculations amounts to 15%. With a decrease in the radius of a cylindrical body, these deviations are reduced.


Introduction
The study of the dynamics of the interaction of extended underground bar structures with soil during the plane wave propagation along their length is mainly considered in a onedimensional statement [1][2][3][4][5][6][7][8]. In this case, the soil is assumed to be undisturbed, which is true in the areas far from sources of excitation, and rods (or rod structures) with external friction are considered. Non-stationary problems of wave propagation in elastic rods with external Coulomb friction are considered in [9][10][11][12][13][14][15][16]. Problems of this kind with the use of nonlinear laws of interaction (the Winkler type model, the Kelvin-Voigt type and the standard linear body type) were studied in [1-2, 4-8, 17-20]. Depending on the physical and geometric properties of underground structures and soils at distances close to the wave source and in some other areas, ignoring the ground motion is considered unreasonable [21][22][23][24]. In one-dimensional statements, this gap is filled by solving two systems of differential equations for the structure and soil (two one-dimensional problems) [1][2], the connection between which is realized using the conditions (models) of interaction [1][2]20]. In these statements, it is assumed that the plane section hypothesis is fulfilled both for the underground structure (construction) and for soil, and the interaction conditions are directly included in the equations of motion, i.e. the force acting on the interaction contact is replaced by a distributed force acting uniformly on the cross-section of the underground structure and soil. Thus, when considering rods with external friction, the external friction force acting on the contact surface is replaced by a body force and is included in the equation of motion [1-8, 12, 22]. As noted in [1], experiments conducted under the supervision of L.V. Nikitin proved the reliability of these assumptions within the elastic model for small radii of the rod with external Coulomb friction. The assumption on the onedimensionality of the ground motion considered unlimited and working in not only compression or tension but also in shear deformation near the interaction contact, which, in turn, propagates in a non-stationary mode with a limited velocity deep into the structure, can be considered doubtful. As a result, in one-dimensional problems of non-stationary interaction of extended underground structures with soil [1][2], in the sections of the structure, the stresses increase manifold (from two to several tens of times) compared to the decreasing load. The increase in stress in [1] is explained by the fact that the shear stresses acting on the contact during the interaction from a passive force turn into an active force accompanying the motion. In [25][26][27][28][29][30][31][32][33], the longitudinal interactions of a solid body with the surrounding soil under the conditions of dry friction interaction are considered. In [1][2][4][5], an increase in stresses in a cylindrical structure is also shown. In order to clarify the causes of such "phenomena", the area of applicability of one-dimensional calculations and the influence of the wave amplitude on the friction force and stress-strain state (SSS), we consider the problem of the longitudinal interaction of a cylindrical body with external Coulomb friction in a two-dimensional statement.

Statement of the problem
Consider an elastic semi-infinite cylinder with radius , surrounded by a rough nondeformable body. The origin of the cylindrical coordinate system is placed in the initial section of the cylinder so that the -axis of symmetry coincides with the axis of the cylinder, is the radial coordinate ( Figure 1).
The cylinder dynamics equations correspond to the following system of equations: (3) where are the particle velocity, and are the components of the stress and strain tensor; is the relative volume; is the initial density of the medium.
The equation of state of the cylinder is taken as elastic and it has the following form: (4) where -modulus of elasticity, -Poisson's ratio. Let a plane wave be formed in the initial section of the cylinder x=0 by a load that varies according to the following law: for (5) where is the maximum value of the load, is the time of its action, -Heaviside function. Before the application of load (5), the cylinder is considered to be at rest and stress-free. From the time the load (5) is applied, a plane longitudinal wave propagates along the axis along the cylindrical body, and the problem is considered axisymmetric. Behind the wave front, i.e. with the excitation of the cylinder, radial and circular stresses arise, and due to the non-deformability of the medium, a friction force appears on the side surface, acting against the movement of the cylinder.
The friction force on the surface of the cylinder is taken in the form of Coulomb dry friction [9][10][11] arising from the amplitude of radial stresses. Then, on the lateral surface of the cylinder in the case of contact, the following condition should be satisfied for : where is the radial stress of the cylinder on the lateral surface, determined in the course of solution. It should be emphasized that in the case of separation of the cylinder from the surrounding medium, i.e., for , the boundary conditions (6)-(7) take the form , .

On the method of solving the problem and approbation of the method
The boundary value problem posed, i.e. the system of equations (1)-(4) with zero initial and (5)-(7) with boundary conditions, is solved numerically -by the method of finite differences, similarly to the problem given in [9][10][11]19]. The finite difference scheme is given in [19,34], and it has the second-order accuracy. Consider the results of numerical solutions to the problem obtained on a computer using the second condition in (7). The algorithm and program for solving the problem were tested by comparing the calculation results with exact solutions. Figure 2 shows the distribution of longitudinal stresses along the length of the cylinder for different points of time in the absence of friction (curves 1, 2) and the results of calculations with Coulomb friction, the friction coefficient of which is 0.1 (curves 3, 4). Calculations were performed using the following characteristics [1,22]: 7800 kg/m 3 , Е=2. 15      The first test example, where and are satisfied at the boundary, is essentially equivalent to the classical problem of the plane longitudinal wave propagation in an elastic half-space [22] and has an exact analytical solution. Solid curves 1, 2 in Figure 2 refer to exact solutions, and dotted curves refer to numerical solutions. Solid curves 3 and 4 refer to approximate solutions of the one-dimensional problem [22]. Of the onedimensional problems [1][2][3][4][5][6][7][8], the most suitable for the problem under consideration is the one given in [1,22], where the friction force is formed from the amplitude of the propagating wave. With Poisson's ratio 0.3 and friction 0.1, the solution satisfies the necessary limitations of the one-dimensional theory with acceptable accuracy [1]. The algorithm and program for solving the problem were also tested by comparing the calculation results with the experimental ones. Figure 3 shows the comparison of the numerical result with the result of L.V. Nikitin's experiment given in [1] and the numerical solution to the one-dimensional problem by the method of characteristics [1]. In [1], the section with friction was located along the steel tube from 1 m to 1.5 m, and the stress pulses in the experiment were measured in front of the section with friction for 0.4 m (Figure 3,a) and behind the area with friction for 1.6 m ( Figure 3b). To obtain a numerical solution in areas without friction and in the inner surface, the free surface conditions were accepted, and in areas with friction, it was necessary to meet conditions (6)- (7) with the addition of a pressing force value of 1.05 MPa to the radial pressure. The parameters of the problem remained the same except for 0.425; 0.0127 m and 0.007 m (diameters of the tubes); Т=0.1 ms. The above comparison shows that the result of the numerical calculation coincides with high accuracy with the results of the experiment, including the smeared wave fronts. Note that the oscillation of the numerical solution behind the wave front is not a drawback (with the exception of smeared fronts) of the method used, but a consequence of ignoring the normal static pressure in the area with friction when applying the initial stress state condition in a two-dimensional statement [10][11]19].
It should be noted that if in the first test example, where shear stresses were practically absent, the introduction of pseudoviscosity in the form (8)  was sufficient (curves 1, 2, Figure 2) for 40, then in the problems under consideration for high friction coefficients, tangential forces arise, especially on the side surface, and the volumetric viscosity (8) turned out to be insufficient. We had to add an artificial viscosity tensor to the stress deviators in the following form [34]: (9) where , 0.2.

Numerical results and their analysis
The numerical results obtained for the problem posed are presented in the form of graphs. A series of calculations were performed for various values of the coefficient of friction, the rod radius, the length of a rectangular pulse in time, the amplitude of the specified load, and the physical and mechanical characteristics of the cylinder (see Table 1). Let us consider changes in the wave parameters in section 0.5 . Figures 4-5 show changes in longitudinal stresses and particle velocity in time for option 3 at fixed points in this section. Curves 1-4 in the figures correspond to the points 0, 0.4 m, 1.2 m and 2 m, respectively. As seen in Figure 4, the amplitude of the maximum longitudinal stresses decreases with distance. Further, at a fixed point, the stress values practically remain constant in time. The farther the distance of the considered point from the cylinder end (load application), the more "strong" absorption of stress values is observed at this point (curves 3-4, Figure 4). The velocities of the particles of the cylinder decrease with distance from the cylinder end, and in time ( Figure 5). Interestingly, the decrease in velocity with distance and time occurs approximately along the same line (curves 1-4, Figure 5). The distributions of longitudinal stresses and velocity along the length of the cylinder for this option at different time points are shown in Figures 6-7 (curves 1-4 correspond to time points 3 , 5 , 7 , and 10 ). Here, the values of longitudinal stresses decrease with distance along one line, which decreases approximately exponentially. The velocities of the particles of the cylinder behind the wave front remain practically unchanged (constant), but here also a decrease in their values with time is observed. The decrease in these parameters is explained by the fact that behind the wave front in the perturbed cylinder, friction forces appear on the outer surface of the cylinder, which are proportional to the emerging radial (normal to the side surface) stresses according to (6) and act against the movement of the cylinder. In the absence of friction force, i.e. at slippery contact, from the test example (curves 1-2, Figure 2), it can be seen that behind the wave front the stress values remain constant, in other words, do not depend on distance and time, as in (5).   The appearance of friction force (6) on the outer surface of the cylinder (at the contact) when a constant load (5) is applied (Figures 4-7) results in a decrease in the amplitude of stresses along the distances independent of time, while the amplitude of particle velocity decreases in both variables.    Table 1 (item 1). From , it can be seen that an increase in the values of the friction coefficient leads to an intensive decrease in the amplitudes of longitudinal stresses.  Changes in elastic energy in a cylinder in time for various values of the friction coefficient and the duration of load acting on the cylinder end are shown in Figure 9. With a constant load acting on the cylinder end under longitudinal wave propagation along the elastic cylinder, the accumulation of elastic energy occurs with time. In the absence of a friction force (option 1), the accumulation of elastic energy occurs linearly. With the appearance of a friction force on the contact surface, part of the energy is spent to overcome the friction force (options 2, 4, 6, Figure 9). In the case of 0.5 (option 6), the loss of elastic energy is the largest, which confirms the result shown in Figure 8. Thus, the decrease in the parameters of waves in the cylinder is explained by the consumption of elastic energy for friction (resistance) of the outer surface of the cylinder. In the case of limiting the time of the load action (5), after the stress is removed, the elastic energy decreases, and not its conservation, obtained under the action of load on the cylinder end (options 7 and 8, Figure 9). Figure 10 shows the distribution of the maximum values of longitudinal stresses along the length of the cylinder for different calculation options, which correspond to the numbers of curves. Here, too, a decrease (approximately by an exponential law) in maximum stresses with distance from the origin of coordinates is observed, as stated in problems for small rod radii and friction coefficients [11]. An increase in the radius of the cylinder 0.4 m (option 9) and 0.6 m (option 11) leads to a "weak" decrease in the values of maximum stresses at the same values of the friction coefficient, i.e. the values of maximum stresses increase compared to option 3 ( Figure 10).
As seen in Figure 5, the decrease in the velocity of particles in the cylinder with distance and time occurs approximately along one line. Figure 11 shows the decrease in the maximum values of particle velocity in the initial section with time. The value of the maximum particle velocity in the cross-section corresponds to the velocity values shown in Figure 11 at . Let us consider cases of decreasing the time of action of the dynamic load on the cylinder end. In options 7 and 8, after relieving the load in section 0, a rarefaction wave propagates along the cylinder (Figure 12), and at the cylinder end, the stresses at turn to zero and this boundary plays the role of a free surface. Therefore, after the load is removed, the particle velocities take on a negative value, and the reverse movement of the cylinder occurs (Figure 13), which also decays in absolute value with time due to the stresses rarefaction in the cylinder. Curves 1-4 in Figures 12-13 correspond to the points shown in Figure 4.  Thus, when a rectangular pulse acts on the end face of a cylinder, the pulse amplitude is depleted with distance, while maintaining its length ( Figure 12). The run of curves 2 and 3 for option 8 in Figure 12 is qualitatively similar to the results obtained in [1]. The quantitative discrepancy is explained by the fact that in the experiment [1] the law of friction is formed by the wave amplitude and pressure (compression) from the outer side of the rod, and by the discrepancy between the accepted geometric dimensions of the rod.  Changes in the physical and mechanical characteristics of the cylinder and the value of the load (5) on the cylinder end did not lead to a qualitative change in the wave parameters shown in Figures 4-11. The two times decrease (option 13) and increase (option 4) of the maximum stress in (5) acting on the cylinder end are similar in quality to the results of calculations of option 3 (Figures 4-7), and in quantity, the values of the parameters decrease (option 13) and increase (option 14) by a factor of two, respectively. Varying the physical and mechanical characteristics of the cylinder material also leads to similar results in quality. In options 15, 16, and 21-23, slight decreases in the stress amplitude and a delay in the arrival of the wave front are observed, therefore, a decrease in the wave propagation velocity. As for the results of calculations of options 17-20, a reverse pattern was obtained, i.e. in terms of quantity, they lead to a slight increase (by 7% compared to options 3 and 6) in the stress amplitude and wave propagation velocity.
We also note that the changes in the wave parameters (longitudinal stresses and velocities) shown in Figures 4-8 longitudinal stresses but in quantity. The explanation for this can be seen from , determined from (6) and shown in Figure 14.  Let us now consider the patterns of change in shear stress (friction force) on the outer (contact) surface of the cylinder ( Figure 14). In fact, shear stresses on the contact surface arise with the appearance of normal (radially tensile) stresses, i.e. with the arrival of a propagating wave and due to the incompressibility of the external medium. The curves in Figure 14 refer to points 0.4 m; 1.2 m and 2 m on the side surface of the cylinder ( ). As we see in Figure 14, with distance, the shear stresses also decrease, and at fixed points behind the wave front, they remain practically constant in time, as do the changes in the pattern of longitudinal stresses in Figure 4. The change in shear stresses in time at the internal points of the cylinder is shown in Figure 15. The appearance of shear stress at the internal points of the cylinder is connected with the propagation of elastic shear (tangential) excitations that occur on the contact surface. In section 0.5 , the shear stresses fluctuate with the stresses on the contact surface of the cylinder. Their fluctuations are explained by numerous reflections along the radius of the cylinder. The smaller the radius of the cylinder, the greater the frequency of their oscillations. Since the radius is relatively small with respect to the length of the cylinder, the acting contact force (friction) on the outer surface at a velocity of at time is distributed over the entire radius of the cylinder. Thus, for an elastic problem with a small radius, the solutions obtained in the twodimensional formulation qualitatively coincide with the solutions of the one-dimensional statement [1][2]. Quantitative differences, as we have already noted, are the result of ignoring the amplitude of the propagating wave in one-dimensional problems and adding external pressure to the friction force [1][2].   Figure 16 shows the changes in longitudinal stresses in time for option 12 in section 0.4 m for different points along the radius ( 0; 0.5 ; ). As we see in Figure 16, for 0.6 m and 0.5, the values of longitudinal stresses slightly decrease with distance from the axis of symmetry, in other words, there is a violation of the plane section hypotheses, taken in one-dimensional calculations. These deviations are up to 8-10%. The values of the stress amplitude decrease with the distance from the axis of the cylinder. The same pattern is observed for particle velocities. Figure 17 shows the particle velocity distributions along the cylinder radius in section 0.5 m at 7 . Curves 1 and 2 correspond to the friction coefficients for options 3 and 5. Here the deviation from a straight line (plane section) is up to 8% for 0.2 and up to 15% for 0.4. In general, judging by the calculations, the error of the result in the initial cross-sections [11] is up to 15%, depending on the accepted values of the friction coefficient.

Conclusion
Thus, a two-dimensional problem of longitudinal wave propagation in an elastic cylindrical body with external Coulomb friction is posed in this article. Since the cylinder is inside an unstrained medium, the friction force arises with the arrival of propagating waves, and its value depends on the SSS of the cylindrical body. The problem posed is solved by the method of finite differences. The solution of test examples and comparison with the experimental results confirm the reliability of numerical calculations. Numerical results, presented in the form of graphs, were analyzed. It was established that the parameters (stresses and strains) of waves propagating in an elastic cylindrical body with external Coulomb dry friction attenuate with distance. The mechanism of reduction of the SSS and wave parameters is explained by the consumption of elastic energy to overcome the friction force on the contact surface. The results obtained for the two-dimensional problem are compared with the results of a similar problem in the one-dimensional theory, where the friction force enters directly into the equations of motion. The deviations of the results of the one-dimensional theory are up to 8-15% depending on the values of the friction coefficient taken in the calculations, i.e. violation of the hypothesis of plane sections in one-dimensional calculations is 15%. It is shown that as the radius of the cylindrical body decreases, the error of the one-dimensional theory decreases as well.