Spread waves in a viscoelastic cylindrical body of a sector cross section with cutouts

In this article an analysis of well-known works related to wave propagation and dispersion dependences in a cylindrical waveguide are presented. A mathematical model, methodology and algorithm for solving the problem of wave propagation in a cylindrical waveguide, having a sector cut are developed. The obtained equations are solved by the orthogonal sweep method in combination with the Mueller and Gauss methods. The dispersion relation for a viscoelastic cylindrical waveguide, having a sector cut in cross section at an arbitrary angle is obtained. Based on the obtained results, it was found that, there are no waves in the elastic cylinder of a sector section with real parts of the phase velocity. It has been established that, in the case of a wedge-shaped viscoelastic cylindrical panel, for each mode, there are limiting wave propagation velocities and they change with a change in the radius of curvature. The spectral sets of normal waves with an increase in the angular parameter of the sector cut-out, corresponding to lower non-zero frequencies of wave locking, slowly decrease.


Introduction
Among the variety of products of the metallurgical, engineering, petroleum and transportation industries, there is a wide range of extended objects, the length of which exceeds the transverse dimensions by more than a hundred times. Such objects include pipes, cylindrical rods or other extended objects having various defects [1][2][3]. Currently, there is a keen interest in non-destructive testing methods for extended objects based on the use of normal waves -waveguide methods [4][5][6]. In a number of works, to control linearly extended objects, it is proposed to use a rod wave in the region of minimum velocity dispersion and torsion wave mode in which there is no dispersion [7,8]. As an informative parameter, when the waveguide control of linearly extended objects, as a rule, the reflection coefficient is used. The specified parameter does not allow to detect longitudinal defects [9,10].
One of the main problem of the dynamic theory of elasticity is the study of the propagation of waves in wedge-shaped bodies (or waveguides) [11][12][13][14]. The main features of the waveguide are the length in one direction, as well as the limitation and localization of the waves in other directions. These waves everywhere, in their characteristics, are similar to Lamb waves.
Dispersion dependences having a certain number of traveling wave modes in the frequency range were obtained in [15][16][17][18].
In [19][20][21][22][23], dynamic behavior and wave effect in mechanical systems were discussed as in the material and paying attention to of the construction features.
These are just some of the works that are devoted to assessing the dynamic behavior and the phenomena of propagation of waves in various structures and systems.
The waveguide has a collinear direction Oz axis. The problem of analyzing the spectra of normal waves along the waveguide under consideration is formulated using the relations of the spatial linear mathematical model of the dynamic state of stress-strain of bodies paying attention to viscoelastic properties. These relations are formulated for projections of the dimensionless vector of dynamic elastic wave displacements on the axis of the cylindrical coordinate system {u r , u θ , u z }, as well as for dimensionless characteristics of the state of stress-strain of the object in question at the main sites of the cylindrical coordinate system {σ rr , σ θθ , σ zz , σ θz , σ rz , σ rθ } [12].
Deformation and stress are related by equality is as follows [19] ik ik ik Here ik  -stress tensor, ik  -tensor strain,  -volumetric deformation,  and  -operator modulus of the Fourier images ( cosine and sine images) of the core relaxation of the material. On the function of influence requirements for continuity (except), monotony, integrability, sign of certainty: Equations of motion of a viscoelastic cylindrical mechanical waveguide, occupying region V, are defined by the following equations [12]: -components of the strain tensor. The above equations (4), (5), (6) form a system differential equations in partial derivatives with complex coefficients. The constructed system of equations is resolved with respect to the first derivatives where is introduced the notations: The boundary conditions are set in the form: . 0 , 0 , :

Solution Methods
The geometry of the object and the natural assumption about the nature of the wave motion along the axis Oz allow us to present the problem's (7) -(9) solution in a form, that allows separation of variables [9,26]    Consider the following two cases: 1) С = С R +iC i , k = k R ; then solution (10) has the form of a sinusoid, amplitude of which is decays; 2) С = С R , k = k R +ik I ; the oscillations are steady, but in z, damp. In further calculations, the index n is skipped. If we substitute (10) into (7), (8), (9), then the constructed system of differential equations in partial derivatives with complex coefficients reduces to a spectral problem.
To describe the wave process, we use the relations (1), (2), (3) given in the previous paragraph. The resolving system of equations coincides with system (7), the boundary conditions on the surface (8) are preserved without change. The boundary conditions for an arbitrary angle of the wedge, in the case of a free lateral surface, should be written in the form: where 0  -angle at the top of the wedge.
Under condition (12), the separation of the variables r and φ is no longer possible; therefore, the direct method is used [29]. In view of (12), the system (7) has the form: The boundary conditions (8) are similarly transformed The components of the tensor of stress   , z   and zz  are expressed through the basic unknowns: Then, paying to attention the first equation of (15), the boundary conditions (14) take the form: The boundary problem for the system (13), (14), (16) can be led to the boundary problem for the system of differential equations, using the direct method, which in the solution will make it possible to use the method of orthogonal sweep [29].
where i varies from 0 to, As a result of sampling, the vector of basic unknowns with a total dimension of 6N can be written as: For variable   conditions (16) paying to attention using the central differences The first and third of conditions (16) paying to attention, approximating the derivatives with respect to υ: Derivatives for the line with the number i =N , paying to attention boundary conditions at The number of straight lines can be halved by using the conditions of anti symmetry of transverse vibrations of the plate at 0   : And, according to (15) we get the system of differential equations , In equations (27), expressions for derivatives w iυ , v iυ , u iυ , σ i,υ , β i,υ , τ i,υ from relations (23) -(26) are selected. Conditions of the free surface are obtained in the form Thus, the spectral problem (13), (14), (15), using the discretization of the coordinate φ by the direct method, reduces to the canonical problem (27), (28), which we solve using the method of orthogonal sweep [30].

Numerical results
As an example of a model of material, we choice the three parametrical relaxation core: . Dimensionless quantities are chosen so, that the shear rate Сs , density ρ , radius R have unit values, and the ratio of Poisson υ=0,25, kernel parameters: Table 1 shows the limiting values of the phase velocity, depending on the angle of the wedge (for the first mode of the edge ). The found phase velocities within the framework of the described methodology for calculating a wedge are given in columns 5-6 for various boundary conditions. In column 5 reprecented results calculation options with three internal lines with the boundary conditions (17) Also, in columns 3 and 4, respectively, the limiting values of the real part of the phase velocity of the first edge mode are given, which was learned in [15] by hypotheses of Kirchhoff -Love and Timoshenko plates. In column 6 shows the calculation results obtained by the formula The phenomenon should be paying to attention, when studying the dynamic behavior of a cylindrical waveguide of variable thickness. In the case of a cylinder with a sector cross-section, the first complex mode has a frequency cutoff. On locking, the axial displacements are equal to zero and the oscillations of the infinitely viscoelastic cylinder with a sector cross-section occur in a flat deformed state. In the second mode, at the locking frequency, are observed, the ring and radial displacements are equal to zero. Unlike edge waves, in a cylinder of infinite length with a sector cross section, in an acute wedge the waves do not have a final solution for.

Conclusions
As a result, the following results were obtained: 1. A mathematical model, technique and algorithm has been developed for solving the problem of propagation of waves in a cylindrical waveguide (viscoelastic), having in the cross section of the body a sector cut.
2. When studying the dispersion dependence for a waveguide with a sector cross-section, it was found that the first complex mode has a frequency cutoff. On locking, the axial displacements are equal to zero and the oscillations of the infinitely viscoelastic cylinder in the sector cross section occur in a flat deformed state.
3. Paying attention to properties of the material makes it possible to evaluate the damping abilities of the system as a whole and reduces the real parts of the wave propagation velocity by 10-15%.
4. It has been established that the spectral sets of normal boundary waves with increasing angular parameter of the sector cut-out, corresponding to the lower non-zero frequency of locking the waves, slowly decrease.