Derivation of the one-dimensional radiation condition in elasticity theory by introducing infinitesimal viscosity

. With the rapid development of engineering science and technology, tasks related to the calculation of wave processes in elastic medium are becoming more and more relevant. One of the important problems is the search for effective methods for calculating the propagation of harmonic oscillations in elastic mediums, in particular, in semi-infinite spaces. One of these methods is the use of asymptotic boundary conditions of the "radiation conditions" type, which allow us to describe the processes of wave propagation outside the computational domain. In this paper we consider the application of infinitesimal viscosity to derive an asymptotic boundary condition of the "radiation condition" type for the propagation of harmonic oscillations in a semi-infinite elastic space. The results of the study can be used in the design of various devices and structures subjected to the influence of wave processes in elastic mediums, in particular, in the field of soil mechanics.


Introduction
In elasticity theory, harmonic oscillations are an important tool for investigating various physical processes related to the determination of the stress-strain state in elastic media during non-stationary but steady-state oscillations.Of particular interest is the study of propagation of harmonic oscillations in semi-infinite elastic media.In considering the behavior of waves in such media, it is often necessary to introduce a boundary condition ensuring that the wave field does not become unbounded as it propagates away from its source.This boundary condition is known as the "radiation condition" and it plays a crucial role in many applications [1][2][3][4].
While the radiation condition is well established for linear wave equations in infinite regions, it becomes more complicated in semi-infinite regions.In these cases, it is not sufficient to impose a boundary condition which simply reduces the wave field to infinity since waves can also bounce back from the boundary resulting in interference and standing waves.Various methods have been proposed over the years to solve this problem, such as perfectly matched layers (or damping interlayers), absorbing boundary conditions, and the use of auxiliary boundary conditions.
In this paper, we propose a new method for obtaining the asymptotic boundary condition of "radiation condition" type for harmonic wave propagation in semi-infinite elastic medium.The condition is derived by introducing an infinitesimal viscosity term into the solving equations which allows us to obtain a solution that satisfies both the solving equations and the boundary condition.The asymptotic boundary condition obtained is valid for both homogeneous and inhomogeneous mediums and can be applied to various wave phenomena such as seismic, acoustic and electromagnetic waves [5][6][7][8][9].
In industrial settings, non-destructive testing is often used to inspect structural components without causing damage.Elastic waves can be used to detect internal defects, cracks in sufficiently long construction elements as well as anomalies in the material.The proposed method for deriving the radiation condition can be used to accurately model and predict the behavior of elastic waves during testing.Acoustic imaging techniques, such as medical ultrasound, use elastic waves to generate images of internal structures.Considered method can improve the accuracy of these imaging techniques by providing a more accurate model of wave propagation in the medium.This can lead to improved medical diagnoses and non-destructive testing.

Methods
Let us briefly summarize the main points of the theory of elasticity.Generalized Hooke's law for a linear elastic medium has the form [10,11]: Equation of motion of the medium in the absence of mass (volume) forces [10,11]: where the dot above the letter means the time derivative.

 
,, , in terms of displacements, the equation of motion in terms of Hooke's law will be as follows [6,7]: When solving a wave equation For the uniqueness of the solution of this equation, 2 boundary conditions are required on both sides: at zero and at infinity.The question arises, which condition should be imposed on infinities.The answer is in the so-called Sommerfeld's "radiation condition" [8][9][10][11], which in one-dimensional, two-dimensional (in cylindrical coordinates) and threedimensional (in spherical coordinates) cases has the form [8][9][10][11] A large number of different mathematical models are used to describe the phenomenon of viscoelasticity.The elements of these models are a piston and a spring.The spring describes a perfectly elastic solid body and obeys Hooke's law.Whereas the piston describes the behaviour of a perfectly viscous body.The simplest of these are described using combinations of these two elements.Examples are the Maxwell (for liquid) (5) [12][13][14][15][16][17]: (5) Kelvin-Voigt (for solid) models ( 6) [12][13][14][15][16][17]: Fig. 2. Standard linear body model.
In what follows, we will consider an approach replacing the elastic medium by a viscoelastic one to determine the asymptotic behavior at infinity by the example of an infinite half-space.

Results
Let the plane =0 z vibrate according to a given harmonic law.Consider the law of oscillations of an infinite half-space filled with a homogeneous and isotropic elastic continuous medium >0 z .In our case we have a one-dimensional problem: the displacement z uu  will be only along the axis z; the x, y variables will not affect anything.Formulas (1)-(3) simplifies: where the dash denotes the derivative on z .As a result, we have: We have obtained a partial differential equation of hyperbolic type.Suppose we need to find its steady-state solution (which does not depend on initial conditions).
Consider a problem statement with specific boundary conditions at zero.If the plane =0 z moves according to a harmonic law, the boundary conditions will take the form: To simplify the mathematical calculations, passing to complex numbers Substituting an established solution in the form of   , = ( ) it u z t Z z e  , we get an equation on the function Z : The general solution to this equation is where: The boundary condition on Z at zero: Consider another case and let stress in =0 z is changing according to a harmonic law: Then the boundary conditions will take the form: Turning again to the complex values   = u Re u , we get: : In this case the boundary condition Z at zero: There is only one boundary condition on function Z .To determine the two constants 1 C and 2 C , we need to implement "radiation condition", which introduced in previous Section (4).
In our one-dimensional case, the boundary condition at infinity (radiation condition) takes the form: This condition is satisfied only by the second solution in Z .Then, considering the boundary condition at zero, we obtain   02 0 = .The radiation condition is derived in the above papers by introducing an effective absorption, obtaining a solution, and then letting the absorption go to zero (principle of ultimate absorption).Indeed, any small absorption is necessary to reduce the influence of initial conditions on the steady-state solution to zero.Let's try to get the same solution by changing the elastic medium to viscoelastic and then pushing the viscosity towards zero.
Let's take the simplest viscoelastic medium model, the Kelvin-Voigt model.Hooke's law is modified by parallel addition of a viscous (Newtonian) element: The final partial differential equation will then be of the third order: A general solution to this equation is: Since the real part of the first root is positive, it corresponds to an infinitely increasing solution at infinity, only the second solution makes physical sense.It is not difficult to see that when η approaches zero, 2 λ ik  and the second solution goes into just ikz e  .In the end, we have seen that the principle of radiation gives the right condition at infinity.

Conclusions
The following conclusions can thus be drawn: 1.By introducing viscosity into the solving equations, one obtains a unambiguous solution since the second, "non-physical" solution is discarded by the boundedness requirement at infinity.Subsequent increasing of the viscosity to zero leads already to a undamped solution at infinity which, as a consequence, is also unique.Thus, the appearance and disappearance of viscosity is a kind of "focus" to separate the "physical" solution from the "non-physical" one.2. This solution has been shown to be fully equivalent to existing methods, in particular the use of the so-called " radiation condition " by Sommerfeld, derived it is sometimes required not to solve a problem with initial conditions (a Cauchy problem), but a steady-state solution defined only by boundary conditions.In this case.This happens, for instance, when external conditions change according to the harmonic law ω it e during a long period of time.Then in the steady state the time-dependence of the solution also becomes harmonic and it is convenient to represent the solution as     ω , , , , , it u x y z t v x y z e .In this case the wave equation turns into the Helmholtz equation:

Fig. 1 .
Fig.1.Kelvin-Voigt and Maxwell model.the standard linear body (7) [12-17] (mathematically equivalent Poitting-Thomson-Ishlinsky and Ziner-Rzhanitsyn models for solid and similar models for liquid): case, considering the boundary condition at zero, we obtain: Conferences 410, 03025 (2023) https://doi.org/10.1051/e3sconf/202341003025FORM 33) Substituting an established solution in the form of   ω , = ( ) it x t Z z e , we get an equation for the function Z : roots.One of the roots has an argument in the interval ππ ;