Wave propagation in a cylindrical body in the presence of surface forces depending on the relative displacement

. Studies of wave propagation in extended bodies with external friction have a wide range of applications and are significant for various engineering and scientific fields. They contribute to the development of new technologies, improve the design and construction of structures, and expand our understanding of the physical processes occurring in various materials and media. In this article, axisymmetric two-dimensional problems of the propagation of longitudinal waves in a cylindrical body are numerically solved in the presence of surface friction forces of the Winkler and Kelvin-Voigt types. For the numerical solution, the Wilkins scheme of the finite difference method was used. The influence of friction forces on the wave parameters is revealed. It is determined that the results of the considered problems are between solutions using slippery contact without friction and with friction according to the Coulomb law. A 5-7% deviation of the hypothesis of flat sections is shown, which makes it possible to reduce such a problem to a one-dimensional formulation.


Introduction
The propagation of waves in a cylindrical body with surface forces depending on the relative displacement is an interesting problem in the field of mechanics of deformable bodies.Such types of problems can arise, for example, in the study of dynamic waves in elastic pipes or rods, when various forces act on their surface, depending on strains or relative displacements.Mathematical modeling of wave propagation in such bodies is generally based on the equations of the dynamics of a deformable body.Problems of an extended body with external friction (dry Coulomb friction; Winkler type friction; friction depending on the time factor, etc.) belong to the class of geo-mechanical problems where the interaction of a rigid body (for example, a rod or a pipe) with soil is studied, taking into account friction on the outer surface of the body.In a simplified one-dimensional formulation of the problem, wave propagation with external friction is considered, i.e. the forces acting on the outer surface of an extended body are included in the equations of motion [1][2][3][4].Many problems [5][6][7][8][9], applied in the field of seismic resistance of underground pipelines, were solved in this formulation, with various types of external frictions.
As is known, the nature of the occurrence of the Coulomb friction force is such that it reaches its limiting value through a certain relative displacement of two media.A pattern of such a change is given in [1,[10][11] and experimental studies of changes in friction forces [12] led to such conclusions.For underground pipelines interacting with soil, friction forces obey the laws proposed in [1,13].The model of interaction given in [13] practically describes the pattern of change in the friction forces, presented in [1], up to reaching the limiting value.Further development of the model of interaction of underground pipelines with soil in [1] contributed to the development of various conditions [13] that take into account such properties of the pipeline and soil as the local character of interaction, structural changes in interacting media, etc. and applied in works [14][15][16][17][18][19][20][21][22][23].The purpose of this study is to determine the influence of the Winkler-type interaction condition on the process of wave propagation in a cylindrical body located in a non-deformable and stationary medium.This problem, considered in a non-one-dimensional formulation, is practically a continuation of the studies given in [24].

Formulation of the problem
Consider an elastic semi-infinite cylinder with radius  0 , surrounded by a rough nondeformable body.The origin of the cylindrical coordinate frame , ,  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.The cylinder dynamics equations correspond to the system of equations of the following form: where  are particle velocities;  is the displacement;  ̂ and  ̂ are the stress and strain tensors;  and  ̂ are the spherical and deviator parts of stresses;  and  are the volume compression and shear moduli; = 0 /  is the relative volume;  is the density of the body;  0 is the initial density;  is the Hamiltonian vector operator;  ̂ is the unit tensor.
Let a plane wave be originated in the initial section of the cylinder =0 by the load that varies according to the following law: where   is the maximum value of the load,  is the time of its action.Before the application of load (3), the cylinder is considered to be at rest and stress-free.From the time of load application (3), a plane longitudinal wave propagates in the cylindrical body along the -axis, and the problem is considered axisymmetric.Behind the wave front, i.e. with the disturbance of the cylinder, radial and circular stresses arise, and due to the nondeformability of the medium, a friction force appears on the side surface, acting against the movement of the cylinder.From the analysis conducted in [24] for the problem posed, the boundary conditions on the contact surface are taken in the form of the Winkler interaction, i.e., the following conditions are set on the boundary  =  0 ,  ≥ 0: where the interaction function   (  ,   ) is determined by the following relation [1,13]: is the coefficient characterizing the rigidity of contact with the external medium;  is the index of the range of change in   ;   =  ̅/ ̅ * is the parameter characterizing the degree of destruction of the contact layer of the cylinder with the external medium;  ̅ is the displacement of a particle of the contact layer of the cylinder relative to the external medium;  ̅ * is the critical value of the relative displacement at which the contact layer of the cylinder with the medium is completely destroyed;  is the coefficient of friction.When the interaction function is constant, the friction law (4) transforms into the integral interaction model proposed in [13].The value of   in ( 4) -( 6) is the normal stress acting on the contact surface.It is determined by the normal dynamic stress arising in the process of disturbance of the cylinder.Ignoring the movement of the external medium and the static pressure, we obtain   =|  | for = 0 .
From ( 4) -( 5), taking into account (6), it follows that  ̅ * = /  (7) and the critical value of the relative displacement is a constant value.Thus, the problem under consideration is reduced to solving systems of equations ( 1) -( 2) with zero initial conditions and with boundary conditions at the end (3) and on the outer surface = 0 (4) -( 5).

Solution method
We solve the problem posed, by the finite difference method according to the Wilkins scheme [25] in the same way as in [24].In [23,26], complete finite-difference relations for such a problem are given.The implementation of the boundary condition is available in [24][25][26].Note that the numerical solutions obtained have the second order of accuracy, provided that the scheme stability condition in the numerical solution is met.

Numerical results and their analysis
Consider the results of calculations of the solutions to this system, taking as initial data the modulus of elasticity of the material =2.1010 5 МPа; lateral expansion coefficient =0.3;initial density of the cylinder  0 =7800 kg/m 3 .As in [24], the amplitude of the specified load in the initial section (3) is   =111.63МPа, and the time of its action  is considered to be 1 ms.
Changes in longitudinal stresses   and longitudinal velocities   over time at fixed points in section =0.50 are shown in Figs.1-2.Curves 1-4 correspond to points =0, 0.4 m, 1.2 m and 2 m, respectively.The value of parameter   in the model of interaction was taken equal to 500 м -1 , in (6) =0.2, and the index of the range of change in   (4) was =0, which corresponds to the linear law  in relation to the normal stress and relative displacement.As seen from Fig. 1, the values of the maximum longitudinal stress practically do not decrease with distance from the initial section at the front.The decrease in stresses occurred in [24] when applying the law of interaction in the form of Coulomb's dry friction.The stress values at fixed points behind the wave front remained unchanged, i.e. permanent over time [24].When applying the friction force in form ( 4), the values of longitudinal stresses at fixed points of the cylinder decrease rapidly with time (curves 2-4, Fig. 1).A further decrease in stresses can be explained as follows: according to the Coulomb law, the shear stresses on the lateral surface, acting against the movement of the cylinder, depend only on the normal (radial) stress, which remains constant behind the wave front, and in (4) these shear stresses also depend on the relative displacement, which increases with time as the particles of the cylinder move.Consequently, the frictional force acting against the motion increases.However, here the increase in shear stress occurs linearly with respect to the continuous relative displacement.Reducing the values of the cylinder stresses in this problem does not mean a decrease in the maximum value.Stresses at fixed points decrease monotonically and, eventually, asymptotically tend to a constant value, which coincides with the stress obtained in the case of Coulomb's dry friction.A similar pattern can be seen for the change in the longitudinal velocities of particles over time (Fig. 2).Here, in contrast to [24], the decrease in velocity for different points does not occur along one line (curves 1-4, Fig. 2).With time, the velocities of the cylinder particles approach the values obtained in [24].
Fig. 3 shows the changes in longitudinal stresses along the length of the cylinder for different points of time.Curves 1-4 refer to time points  =  0 / 1 , =5, 10, 15 and 25, respectively.Fig. 4 shows the distribution of longitudinal velocities along the length of the cylinder for the same points of time.Here, too, the difference between the results obtained and the ones given in [24] is observed.The course of dependence   () for different points of time, which have a damping character, does not occur along a single line, as was the case with the Coulomb friction [24].The particle velocity distribution becomes nonlinear with distance from the origin (Fig. 4).It can be seen from Figs. 3-4 that the wave parameters at the front do not decay if condition (4) is met.In general, for this problem, it is also possible to obtain similar dependencies of changes in the wave parameters over time and coordinates given in [24].When applying the law of friction (4), it is confirmed that the farther the distance from the initial section of the cylindrical body, the greater the attenuation of the wave parameters along the length of the cylinder until condition (5) is satisfied.A decrease in stress and velocity (Figs.1-4) is accompanied by normal stresses   =|  | and relative displacements  ̅, which lead to attenuation with distance of shear stress on the side surface of the cylinder (Fig. 1).Figs. 5-6 shows the changes in shear stress and displacement of particles, i.e., relative displacements over time at the contact points of the cylinder.The curve with sign " ' " corresponds to =0.3, and the curves with sign " * " correspond to =2 and =0.2.All curves numbered 1-3 in Figs.5-6 correspond to points =0.4 m, 1.2 m and 2 m in = 0 Tangential stresses for =0 on the contact, developing smoothly (monotonically) reach their limiting value.Compared to Fig. 1, the values of shear stress at any value of the wave amplitude instantly reach the limiting value.In this case, only curves 1, 1*, and 2 reach the limiting value (Fig. 5).For other points, the increment of the relative displacement tends to zero over time.Note that for the calculation for value =2 the shear stress (after certain time) passes through its "peak" value (curve 1* Fig. 5), which is allowed by the interaction model (4).As seen from Fig. 6, the accepted value of parameter  significantly affects the displacement of the particles of the cylinder.An increase in this parameter or the interaction coefficient, or the coefficient of friction leads to a decrease in displacements of the cylinder particles.The change in the parameters of the interaction model (4) does not lead to significant qualitative changes.In quantitative terms, there is an insignificant difference from the results of the problem given in [24].A decrease in the contact stiffness coefficient   leads to a "weaker" decrease in the wave parameters, and as   →0 in (6), hence in (4)  →0, and the solution to the problem coincides with the solution for a slippery contact without friction.An increase in the indices  or   leads to an increase in shear stress and, consequently, to a more intense depletion of the wave parameters.Evidence of the above can be the pattern of stress changes at the considered points of the cylinder over time for =2,   =500 m -1 (solid curves) and =2,   =100 m -1 (dashed curves), shown in Fig. 7.As   → ∞, the solution coincides with the results given in [24].
Varying the parameters that initiate the wave and the physical and mechanical characteristics of the cylinder material leads to the same results that were obtained in [24] with qualitative (Figs.1-7) and (slightly) quantitative changes.The results obtained for sections =0.50 (Figs.1-4) are valid for the results of sections =0 and = 0 with an error of  5 %.
In general, from the obtained solutions, we can conclude that the results of calculations using the interaction model ( 4)-( 5) are between solutions using slippery contact without friction and with friction according to the Coulomb law.

The case of friction of the Kelvin-Voigt type
Let us now consider the problem of wave propagation in a cylindrical body with external friction, which takes into account the influence of the velocity of interaction between the cylinder and the medium.In this case, the friction force on the contact is determined by the interaction law of the Kelvin-Voigt type [13]: where   is coefficient of shear viscosity of the contact layer of the cylinder with the external medium, which has the following form [1]: For   =0, the law (8) transforms into the nonlinear Winkler-type friction law considered in (4).From ( 8)-( 9) for  ̅= ̅ * taking into account (10), we obtain the following relation for the critical value of the relative displacement: where  ̅ * in ( 8)-( 9) is a variable, in contrast to law (4)-( 5).Thus, the problem under consideration is reduced to solving the system of equations ( 1)-( 2) with boundary conditions (3), ( 8)-( 9) and with zero initial conditions.
We consider the results of the calculations leaving the mechanical characteristics of the cylinder and the parameters of the acting load unchanged.Figs.8-9 shows the dependences of shear stress and relative displacement in time on the contact (outer) surface of the cylinder at fixed points along the length.The numbers of the curves in Figs.8-9 correspond to the points =0.4 m; 1.2 m and 2 m.
Solid curves refer to initial data:   =500 m -1 , =0, =1.2, =0.2;dotted curves to =2, =1.2; and dashed curves to =2, =0.The course of curves () in the model of interaction of the Kelvin-Voigt type is approximately similar in quality to dependence () from ( 4) at a constant load on the end of the cylinder.At a low interaction rate, i.e. as  ̅/0,  ̅ * , which is considered a variable (11), also tends to the critical displacement constant determined from (7).Consequently, the result (curves 2-3) practically coincides with the results of the problem considered with Winkler type friction, regardless of the values of the coefficient of shear viscosity of the external medium.An increase in the interaction rate leads to intensive development of shear stress on the contact surface (curves 1) up to the limiting value.Transition to the second stage of interaction (9), i.e. reaching the limiting value of the shear stress (Fig. 8) depending on the interaction rate always occurs earlier than in the Winkler type model ( 4) -( 5).In the case of  ̅/∞ on the contact surface of a cylindrical body, the critical value of the relative displacement tends to zero and the shear stress instantly reaches its limiting value.As  ̅/∞, the friction force on the surface of the cylinder passes to the friction law (9), i.e.Coulomb friction with a constant coefficient .Thus, the second term in model ( 8) determines the rate (criteria) of transition to relations (9), where the shear stress does not depend on the relative displacement.As seen in Fig. 9, changes in the relative displacement are significantly affected by the parameter characterizing the degree of change in the interaction coefficient, also shown in Fig. 6.Changes in the wave parameters in time and coordinate practically coincide with the results of problems with friction (4)- (5).Fig. 10 shows the changes in the longitudinal stress of a cylindrical body over time at fixed points of section =0.50 (=0, 0.4 m, 1.2 m and 2 m, respectively, (curves 1-4).Fig. 11 shows dependence   () at the same points.
Here the solid curves refer to the initial data of the model parameter (8):   =500 m -1 , =2, =1.2, =0.2; and dotted lines refer to the case when only the interaction coefficient is changed -  =100 m -1 .The pattern of changes in these parameters is similar to the results of the previous problem (see Figs. 1-2).The only difference here is that in the results of the problem under consideration, the attenuation of wave parameters occurs slightly intensively behind the wave front due to the second term in (8) until the transition of the shear stress to the limiting value.An increase in shear viscosity also leads to a "weak" decrease in the values of stresses and particle velocity.In general, according to the results of calculating problems using models ( 4) and ( 8), as in one-dimensional formulations [1], the results differ slightly, mainly at low values of the velocity of transverse wave propagation or in an increase in the exponent in (10).In other cases, the results are almost the same.We also note that the values of the wave parameters along the radius of a cylindrical body differ slightly -within the formulation and the initial data obtained; the maximum deviation is  5-7 %.These deviations occur similarly [24].

Conclusion
The results of calculations of wave propagation with external friction according to the Winkler type or the Kelvin-Voigt type interaction models with the transition to Coulomb friction are between solutions using slippery contact without friction and with friction according to the Coulomb law.The values of the wave parameters along the radius of the cylindrical body, i.e. compliance with the fulfillment of the hypotheses of flat sections for one-dimensional problems differ slightly -within the statement and the accepted initial data, the maximum deviation is  5-7 %.

Fig. 1 .
Fig. 1.Change in longitudinal stresses over time at fixed points of the cylinder.

Fig. 2 .
Fig. 2. Change of particle velocity over time at fixed points of the cylinder.

Fig. 3 .
Fig. 3. Distributions of longitudinal stresses along the length of the cylinder.

Fig. 4 .
Fig. 4. Distributions of particle velocity along the length of the cylinder.

Fig. 5 .
Fig. 5. Change in shear stress over time at fixed points of the lateral surface of the cylinder.

Fig. 6 .
Fig. 6.Change of displacement of particles over time at fixed points of the lateral surface of the cylinder.

Fig. 8 .
Fig. 8. Change in shear stress over time at fixed points of the lateral surface of the cylinder.

Fig. 9 .
Fig. 9. Change of displacement of particles over time at the lateral surface of the cylinder.

E3SFig. 10 .
Fig. 10.Change in longitudinal stresses over time at fixed points of the cylinder.

Fig. 11 .
Fig. 11.Change of particle velocity over time at fixed points of the cylinder.