Propagation of longitudinal waves in a linear viscoelastic medium

. The article is devoted to the study of longitudinal wave propagation in a viscoelastic medium. A mathematical model of the problem of a linear viscoelastic medium was developed. The solution to the considered problems is reduced to the solution of a system of differential equations solved by the method of characteristics with the appropriate boundary conditions. The results are compared with known results obtained by other authors; a comparison shows the adequacy of the task posed. It was determined that the maximum values of stress, strain, and velocity of particles in viscous media vary according to a non-linear law. In the initial section of the medium and near it, first, the stress reaches its maximum, and then the strain and velocity of particles reach their maximum values.


Introduction
Many researchers study the solutions to wave problems in soils: Kh.A. Rakhmatulin, S.S. Grigoryan, G.M. Lyakhov, K.S. Sultanov, and others.Special attention is paid to the study of wave propagation in soils.In [1,2], experimental and theoretical interactions of a rigid body with soil are studied.Based on the experimental results, the laws of interaction of underground pipelines with soils were constructed.Numerical solutions to the problem of plane wave propagation in soil are given, taking into account the elastic, viscous, and plastic properties of the underground structure and soil.The results of experimental and theoretical studies on the longitudinal interaction of elastic and inelastic seismic waves with extended underground structures and a barrier in soil media are also presented.As part of the study of waves in soils, the dynamics of shock wave propagation was considered in [3].In [4], based on the analysis of the results of the numerical solution to the wave problem, the conditions for quasi-static soil strain under dynamic loading in the experiment were obtained.By comparing the results of the experiment and numerical calculations and using successive approximations, the refined values of the mechanical characteristics of loess soils were determined based on the elastic-viscous-plastic model of soil strain.In [5], the mechanical characteristics of soil under static and dynamic strains were experimentally determined.The propagation of plane waves in soil was considered.To describe the dynamic strain of soil, the G. M. Lyakhov elastic-viscous-plastic model was accepted.The system of differential equations is solved by the method of characteristics and the method of finite differences in an implicit scheme.The influence of the thickness of the layer on wave parameters and on the quasi-static process is experimentally shown.The quantitative and qualitative influence of the mechanical characteristics of soil on wave parameters is determined.In [6], a one-dimensional statement of the non-stationary wave problem of the propagation and reflection of longitudinal monochromatic waves from a rigid stationary barrier, to which an underground pipeline adjoins, is given.The linear viscoelastic Eyring model, which describes limited creep and relaxation, is taken as the pipeline deformation law.The Eyring model makes it possible to describe the behavior of underground steel and polymer pipelines under dynamic loading.The problem is solved numerically using the theory of characteristics followed by the finite difference method in an implicit scheme.Analysis of the plane wave: longitudinal stress shows that at high frequencies of the dynamic load that initiates the wave, the stress amplitude in the pipeline increases by two or more times compared to the load amplitude.This is due to the superposition of incident and reflected waves in the pipeline and the high rate of pipeline loading.At low frequencies of dynamic loading, such an increase is not observed due to the slow rate of loading.It is shown that the greatest longitudinal stresses in an underground pipeline arise at points close to its connection with a rigid immovable solid body.In this case, the stress amplitude in the pipeline increases by two or more times compared to the load amplitude.
In [7], it was shown that the interaction forces on the contact surface of an underground pipeline with soil are complex and depend on the wave parameters in soil and in the pipeline.Based on the results of the numerical solution to one-dimensional unsteady problems of coupled waves for the interacting system "underground pipeline-soil", the patterns of changes in the interaction forces on the pipeline-soil contact surface were determined.It turned out that an account for the static and dynamic pressure of soil along the normal line to the contact surface leads to quantitatively different results from the ones when only the static pressure is considered.The necessity to take into account the dynamic stress-strain state of soil in the vicinity of the pipeline is shown.The patterns of change in the interaction forces on the surface of contact of the pipeline with soil are determined.
A wave problem posed corresponds to the setting of experiments on the device of dynamic loading of soil.Here, the soil deformation law is assumed to be elastic-viscousplastic.The numerical solution to the wave equations in these problems was obtained by the finite difference method.Based on the analysis of the stress-strain state of soils in various sections obtained by numerical calculations, a condition was derived under which the influence of wave processes on the mechanical characteristics of soils was eliminated.
It was shown in [9] that the experimental diagrams of the stress and strain components for soft soils are non-linear.Nonlinear diagrams qualitatively differ for undisturbed and disturbed soils.Manifestations of non-linear soil properties are related to micro-destruction of the soil structure under compression and, consequently, with a change in its mechanical characteristics -under strain.It was determined that in the process of soil strain, elastic modulus, Poisson's ratio, viscosity, and other mechanical parameters are variables.From experimental results [9] available in scientific literature, changes in the modulus of elasticity and plasticity of soil are determined depending on the values of the compressive strain.It was stated that the strain modulus of clay and loess soils under static and dynamic deformation change depending on the deformation rate, the state of the structure, and the level of compressive load.
In [10], in numerical seismic modeling of wave propagation, the importance of the consideration of surface topography and attenuation was shown, both of which have a great influence on seismicity.In [10], an approach to modeling a seismic wave field in two-dimensional viscoelastic isotropic media with an irregular free surface was developed.Based on the mesh matching boundary method, the second-order two-dimensional viscoelasticity equations in the time domain and the irregular free surface boundary conditions were transferred from the Cartesian coordinate system to the curvilinear coordinate system.To discretize the equations of viscoelastic waves and an irregular free surface in a curvilinear coordinate system, finite-difference operators with second-order accuracy were used.The boundary condition of a perfectly matched convolution layer was chosen to effectively suppress artificial reflections from the edges of the model.Images and seismograms of the results of numerical tests show that the algorithm successfully models seismic wave fields in viscoelastic isotropic media with an uneven free surface.
In [11], the solution to problems of longitudinal vibration of an underground pipeline was considered, taking into account the linear and nonlinear viscous-elastic-plastic interaction in the "pipe-soil" system under the boundary conditions of rigid fixation or free displacement.The patterns of change in displacements and stresses in pipeline sections under seismic impact in the form of a sinusoidal wave running along the pipe axis were determined.
In [12], a new relaxation model was developed that took into account the Kelvin-Voigt non-local fields of viscoelastic materials and assumed various non-local damping of the elastic and viscous properties of the material.A new relaxation model was realized to study the influence of nonlocal interactions on wave propagation in viscoelastic media and to determine how the contrast between longitudinal and transverse nonlocal fields contributed to the dispersion of propagating waves.Numerical results show two mechanisms of damping of viscoelastic waves, since in addition to viscosity, which is an explicit damping of waves, viscoelastic waves also implicitly damp due to a non-local effect.
A fractional model of a viscoelastic body of distributed order was used in [13] to describe the propagation of waves in infinite media.The existence and uniqueness of the fundamental solution to the generalized Cauchy problem were determined in explicit form.It was determined that the velocity of wave propagation is related to the properties of the material at the initial point of time.Fundamental solutions were also obtained, corresponding to four thermodynamically acceptable classes of fractional linear constitutive models and models of distributed order of power mode.
In [14], the theory of damping and the development of wave modeling with high temporal accuracy were considered, and the complex frequency-dependent velocity was introduced to derive new equations for viscoelastic waves with uncoupled amplitude dissipation and phase dispersion.To obtain high temporal accuracy in viscoelastic wave simulations, a normalized pseudo-Laplacian was used to compensate for time dispersion errors caused by second-order finite difference sampling in the time domain.This greatly reduced the time of low-rank decomposition, Fourier transforms, and greatly improved the efficiency of calculations.It was shown that the proposed scheme could effectively compensate for time dispersion errors and help generate solutions for viscoelastic waves with high temporal accuracy.
It was shown in [15] that seismic modeling of massive structures requires special care since the effects of wave propagation significantly affect the structure response.This becomes more important when the path-dependent behavior of the material is considered.A finite element model is presented for complex non-linear seismic modeling of concrete gravity dams, including real soil-structure interaction.An accurate representation of radiation damping in a half-space medium and the effects of wave propagation in a massive foundation is verified by an analytical solution of vertically propagating transverse waves in the viscoelastic region of a half-space.A rigorous nonlinear finite element model requires an accurate description of the material response.
Aspects of the stability of the walls of the Muruntau open pit and the complex of the KNK-270 steeply inclined conveyor were considered in [16] under the interaction of possible seismic and dynamic loads that arise in mining in the open pit.The main characteristics of dynamic loads were determined on the basis of the obtained records of seismic signal detectors.The amplitudes and frequencies of oscillations of explosive and technological loads affecting the eastern end of the quarry, where the KNK-270 conveyor is located, are determined.The parameters of possible seismic loads were also determined using micro-seismic zoning methods.It was substantiated that seismic loads are the most significant dynamic loads.Amplitudes of technological loads are insignificant and constant.However, they can cause the inclined conveyor to vibrate in dangerous directions.Therefore, when calculating the stability of the eastern edge, it is necessary to take into account all possible types of dynamic loads.
In [17], a mathematical model was developed using a variational approach to study the propagation of vibration waves initiated by railway transport over various distances.A technique was developed for solving the problems under consideration using the finite element method.The influence of the level of vibrations propagating from the passage of railway transport on buildings located at a certain distance from the source of vibration, when the railway track is located at the level of foundation or at a certain height from the foundation, was studied.It was determined that if the railway track is located at a height of 2 m from the level of the ground base of the building, then the amplitude of vibration displacement in the building can be reduced from 1.5 to 3.5 times.
In [18], a mathematical model is presented for determining the dynamic behavior of earth dams, taking into account the viscoelastic properties of soil, using the hereditary Boltzmann-Volterra theory with the A.R. Rzhanitsyn's kernel under periodic kinematic influences.
In [19,20], a detailed review of known publications devoted to the study of the stress state and dynamic behavior of earth dams is presented, taking into account the elastic and elastoplastic properties of soil.A mathematical model was developed using the principle of virtual displacements to assess the stress-strain state of earth dams, considering the elasticplastic properties of soil under static loads.A technique, algorithm and computer programs were developed for estimating the stress state of dams by the finite element method and the method of variable elasticity parameters.A test problem was solved to assess the adequacy of the developed models and the reliability of the technique, algorithms and computer programs.Using the developed models and calculation methods, the stress-strain state of the Pskem earth dam 195 m high was studied, taking into account the elastic-plastic properties.

Statement of the problem and method of its solution
When solving problems of wave propagation in soils, the method of characteristics is used [3].To use this method, we introduce Lagrangian variables as variables.The characteristic method allows us to determine the exact values of the step stress and other parameters.When solving problems on the propagation of plane waves in an elastoplastic medium, modern computational methods are used.In this case, using Lagrangian variables, we take the equation of motion behind the wave front as: The equation of the medium (2.2) depending on function f can be hyperbolic or non-hyperbolic.The problem under consideration is reduced to a hyperbolic form and the characteristic method is used for its solution.If a shock load is created in some section of a viscoplastic medium, then after the step stress, it is found that the stress continuously decreases.The stress behind the wave front is not known in advance, it will be determined when solving the problem.System (2.1), which determines the behavior of the medium, is closed by the following equations: , then the function of characteristics is called the characteristic equation of systems (2.1), (2.3).Here the notion of characteristics is the same as with weak singularities.
If sound waves propagate in the medium, then, at the front,  and  change to infinity.However, it is assumed that at the wave front their differentials are limited.At the wave front, the derivatives have weak discontinuities.
The determination of small changes is reduced to the Cauchy problem and is formulated as follows.Let ht be linear in the plane.Then the values of u , It should be noted that the solution to the Cauchy problem is possible in the case when the stress, displacement, and strain around line   t h are continuous.If they have sufficiently weak discontinuities, then the Cauchy problem is not solved.In the vicinity of line   t h the following conditions must be satisfied: . To solve six equations with six unknowns, the determinants of the systems of equations (2.1), (2.2), and (2.3) must be zero.In this case, As seen from (2.5), there are three characteristics in plane   Under this condition, all three solutions are bounded.To do this, in determinant (2.5) we successively replace the columns and equate each determinant to zero.
Substituting the values of h  from expression (2.6) into (2.7),we obtain the following relations for the characteristics: The value of 1 c varies from particle to particle, so characteristics ht are curvilinear.If the dynamic compression diagram of the medium is linear, then the characteristics are straight lines since const c  1 .This is consistent with the accepted model.
The propagation of a stationary plane one-dimensional wave initiated by a load in some section in a linear viscous medium that increases abruptly and then maintains a constant value was considered in [10,11,12].At that, in all sections of the medium, the stress at the wave propagation by steps or gradually, increases to the same maximum value, and then maintains this value.
The behavior of a viscoelastic medium with increasing and decreasing load is determined by the same equation (2.3).The wave is initiated by a load in the initial section of the medium, which increases at (2.9) Here With these variables, the basic equations of motion take the following form: Load in the initial section in dimensionless form is: The equation of the front line is   x .The following conditions are satisfied at the front: In the new variables, the problem contains two independent parameters S D E E / and  .Reducing the number of parameters allows us to move from one medium and load to another medium and load.For viscoelastic media, an equation of the form (2.7) in dimensionless quantities passes to the following characteristics: Solution in dimensional form is: where max  is the maximum stress for 0  x .

Calculation results and their analysis
The results of calculating the wave parameters at fixed points of the medium are shown in Figs.1-3., the values of the step at the same distances are equal.The step is followed by a continuous increase in the parameters to a maximum, and then their decrease.With distance from the initial section, the maximum values of all parameters decrease.With an increase in  the intensity of the decrease becomes less, and the time to reach the maximum increases.In the initial section and near it, first, the stress reaches the maximum, and then the strain and velocity of the particles.With distance from the initial section, the maximum of all parameters is reached, and at a sufficient distance, over time, almost the same strain tends to a constant value.

Conclusions
1.A mathematical model and method for solving the problem of longitudinal wave propagation in a linear viscoelastic medium was developed using the method of characteristics.2. The maximum values of stress, strain, and velocity of particles in viscous media vary according to a non-linear law. the values of the step at the same distances are equal.After the step, there is a continuous increase in the parameters to a maximum, and then it decreases.5.In the initial section and near it, first, the stress reaches its maximum, and then the strain and velocity of the particles reach their maximum values.6. Strain, at a sufficient distance from the initial section, tends to a constant value over time.

1  1 
, and  are determined from systems (2.1) and (2.3), and the conditions on this straight line are satisfied.If the properties u , , and  on line   t h are continuous, then the Cauchy problem can be solved.If their values are weakly continuous, then the derivatives of these values have discontinuities, and therefore the Cauchy problem cannot be solved.These conditions are always satisfied along line   t h .

..
0  t , by steps up to max  , and then changes according to the following equations: E3S Web of Conferences 458, 08013 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345808013In front of the wave front, the undisturbed medium has the following parametersThe equation of the front line is At h  .The problem is reduced to integration (2.1) closed by equation (2.3that determine the load in the initial section, max  and  .Let us introduce dimensionless Lagrange variables and dimensionless quantities

14 )
E3S Web of Conferences 458, 08013 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345808013where the front can be determined without solving the problem.With (2.12), the first equation is written in the following form: the dependence of the stress at the wave front on time or on distance: Figure 1 refers to stress, Fig. 2 -to strain, and Fig. 3 -to the velocity of the particles.In each case, the magnitude of the step does not depend on parameter  .It decreases with distance.In a viscoelastic medium, in the case when loading and unloading occur according to the same equation,

3 .
The maximum values of strain and velocity of particles change significantly with  .4. It was determined that in a viscoelastic medium in the case when loading and unloading occur according to the same equation,