The propagation of longitudinal stress waves in an infinite rod with a viscoelastic region

The problem of propagation of longitudinal stress waves in an infinite piece-wise homogeneous rod with a viscoelastic region of finite size is solved. A mathematical formulation of the problem and its analytical solution in stresses are obtained. The solution uses the method of the Laplace integral transformation with respect to time. The expressions obtained allow one to determine the stresses in an arbitrary cross-section of the rod at any time. The structure of the solution reflects the process of transformation of the incoming wave with its multiple refraction and reflection at the boundaries of the sections. The solution obtained makes it possible to quantitatively investigate the damping effect of the viscoelastic region and can serve as the basis for selecting the parameters of the insert material when it is used to extinguish the dynamic effects.


Introduction
The use of viscoelastic materials as structural elements, details of mechanisms, and machines is becoming increasingly widespread in such areas as construction (industrial, civil, transportation), engineering, instrumentation, aviation, and many others.This is facilitated, in particular, by the development of technologies for obtaining and practicing the use of polymeric materials, in which the viscoelastic properties are manifested to the greatest extent.Therefore, the study of the behavior of structural elements containing such materials under different types of loading does not cease to be relevant.When it comes to dynamic effects, viscoelastic properties show even those materials that can usually be considered elastic.
A large number of studies have been devoted to the investigation of the behavior of viscoelastic materials and the determination of their properties [1][2][3][4][5].It is shown that the linear theory of viscoelasticity [6,7], based on the mathematical apparatus of Boltzmann-Volter, can be applied to most polymers with sufficient accuracy.
In [17], the problem of the propagation of a shock pulse in a semi-infinite rod consisting of two sections is solved, one of which (finite) has viscoelastic properties.
In this paper we obtain an analytic solution of the problem of the propagation of stress waves in an infinite rod consisting of three sections.The middle portion serves as a viscoelastic insert between two elastic regions.

Problem Statement
The material parameters of the two elastic semi-infinite sections of the rod (x <0), (x> 1) are assumed to be identical and denoted by the index "1"; the parameters of the viscoelastic insert material 0 ≤  ≤  are denoted by the index "2." The longitudinal stress wave ( ) the boundary (x = 0) along the semi-infinite section of the rod on the left.

Research Questions
• How does the process of multiple refraction and reflection of stress waves occur at the boundaries of the rod sections?
• How is the incoming wave transformed as it passes through the viscoelastic region?

Purpose of the Study
• Obtaining analytical expressions for stresses in an arbitrary cross-section of each of the three sections of the rod.
• Analysis of the solution obtained.

Mathematical formulation of the problem
Let us first imagine the stress-strain state of the first semi-infinite section (x<0) as a result of superimposing a given wave and waves reflected from the boundaries (x<0), (x>1): where ( ) ( )  is movements and stresses caused by reflected waves.Then the problem of the propagation of waves in sections of the rod reduces to a system of equations with boundary conditions The initial conditions are zero.The core of the viscoelastic operator has the form [18]:

Solving the problem in images
Applying the Laplace integral transformation to the system (1) with respect to t for the transformed displacements where ( ) ( ) The relation 12 SS can be represented in the form ( ) ( ) Substituting expressions (7) in the general solution (5), we obtain for the transformed displacements:  -the integer part of the number  .

Findings
• Analytical expressions allowing to calculate the stresses in all three sections of the rod at any time and in an arbitrary section are obtained.
• The solution for each section is the result of multiple refraction and reflection of waves at the boundaries of the sections.
• The viscoelastic insert has a damping effect: with each passage through it, the wavelength increases, and the amplitude decreases.

Fig. 1 .
Fig. 1.Infinite rod of constant cross section, having a viscoelastic insert of finite length.
MATEC Web of Conferences 212, 01031 (2018) https://doi.org/10.1051/matecconf/201821201031ICRE 2018 velocity in the i-th part of the rod, and density of the i-th part of the rod,

5 . 3
Finding the originals of stressesTurning on p[20] the expressions for the stresses(7) and noticing in advance that for wave processes, i.e. when p →,