Open Journal of Acoustics
Vol.3 No.3(2013), Article ID:36740,5 pages DOI:10.4236/oja.2013.33011
Against Phase Veloсities of Elastic Waves in Thin Transversely Isotropic Cylindrical Shell
St. Prtersburg State Marine Technical University, St. Petersburg, Russia
Email: alexalex2@yandex.ru
Copyright © 2013 Alexander Kleshchev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received July 18, 2013; revised August 18, 2013; accepted August 25, 2013
Keywords: theory of elasticity; phase velocity; transversely isotropic medium; characteristic equation
ABSTRACT
This paper receives the characteristic equation for the determine of wave numbers of phase velocities of elastic waves, in the thin cylindrical shell with the help of the dynamic theory of the elasticity for the transversely isotropic medium and of the hypothesis of thin shells.
1. Introduction
Based on the use of the dynamic theory of the elasticity for the anisotropic medium and with the help of the hypothesis of thin shells, this paper is determined by the characteristic equation for wave numbers of elastic waves in the thin transversely isotropic cylindrical shell.
2. The Dynamic Theory of the Elasticity for the Transversely Isotropic Medium
Let’s consider the infinite thin transversely isotropic cylindrical shell. The elastic wave is spread along the axis that orthogonal of the plane of the isotropy. The transversely isotropic elastic medium is characterized by five elastic moduluses [1]: or by technical moduluses In the chosen orientation of the axis is the Joung’s modulus, is the shear modulus, is the Poisson’s ratio in the plane of the isotropy. and are the same values in the transverse plane. These moduluses connected with each other by the relationship [14]:
(1)
The Hooke’s law for the transversely isotropic elastic medium is written in the next form [1]:
(2)
where are components of the tensor of deformations, which are equals [1]:
(3)
where are components of the displacement vector
Equations of the dynamic balance in the circular cylindrical system of coordinates [with the harmonic dependence from the time] have the following appearance [14]:
(4)
where
(5)
Components of the displacement vector can be presented in the series form [24]:
(6)
where is the wave number of the elastic wave.
Then we substituted (4) in (5), we receive equations of the dynamic balance in displacements [24]:
(7)
(8)
(9)
where
Now if components of the displacement vector taken from (6) substitute in (7)(9), then we receive following equations for radial functions [24]:
(10)
(11)
(12)
Boundary conditions: normal and tangent stresses are equal zero at external and internal surfaces of the elastic shell are added to equations (10)(12) [24]:
(13)
(14)
(15)
where
3. Hypothesis of Thin Shells
The fellow parameter
can be used for thin shells, where
is middle radius and is the coordinate taking from the middle surface [25]:
(16)
We substitute decompositions in boundary Conditions (13)(15) and 6 equations relative to unknown coefficients [24]:
(17)
(18)
(19)
(20)
(21)
(22)
The rest of equations can be received, by substitution of decompositions (16) in equations (10)(12) and by equated of coefficients at identical powers [24]:
(23)
(24)
(25)
where .
It is necessary to use of equations (23)(25) and for and coefficients with negative indexes are equal to zero. Then in common with the equations (17)(22) the homogeneous system of equations relative to coefficients is formed. Afterwards, we expand the determinant of this system and let this determinant is equal zero we receive the characteristic equation for wave numbers of elastic waves of the mode in the transversely isotropic cylindrical shell.
Now we sell pay attention to elastic waves, which have axial symmetry: the dependence from the angle disappears. If vector of the shell displacement has not of the component, then we have waves with the vertical polarization. In thin case components of strains and tangent stresses are equal to zero, but stresses and are equal [24]:
(26)
(27)
(28)
(29)
Equations of the dynamic balance (their only 2) have the following form [24]:
(30)
(31)
Displacements and can be taken in the form [24]:
(32)
(33)
For the thin shell and can be expanded in serieses:
(34)
(35)
Boundary conditions (their only 2) can be expressed as [24]:
(36)
(37)
The substitution (32), (33) and (34), (35) into boundary conditions (36), (37) and into equations of the dynamic balance (30), (31) results in the system of equations to calculate unknown coefficients The characteristic equation for wave numbers of elastic axisymmetrical waves in the transversely isotropic cylindrical shell we receive by expanding the determinant, which is equals zero. The axisymmetrical wave of the horizontal polarization (torsional wave) has only one component of the displacement vector The problem in this case has the analytic solution. Components of strains are equal to zero, but components of strains and are equal to:
The equation of the dynamic balance has the following form:
(38)
Used (2) and (3), we can describe (38) in the form:
(39)
The component U_{φ}_{ }can be presented as:
(40)
where is the torsional wave number.
We substitute (40) in (39) and have:
(41)
The Equation (41) is the Bessel’s equation for Bessel’s and Neiman functions of the first order:
(42)
where and are arbitrary constants;
From the boundary condition, we receive the characteristic equation for torsional wave numbers:

where
.
4. Conclusions
In the paper, we found the characteristic equation for wave numbers of elastic waves in thin transversely isotropic cylindrical shell with the help of the dynamic theory of the elasticity for the orthotropic medium and of the hypothesis of thin shells both for three—dimensional and axially symmetric problems.
5. Acknowledgments
The work was supported as part of research under State Contract no. P242 of April 21, 2010, within the Federal Target Program “Scientific and scientific—pedagogical personnel of innovative Russia for the 20092013”.
REFERENCES
 S. G. Lekhnitsky, “Theory of Elasticity of an Anisotropic Elastic Body,” M.: Science, 1977, p. 416.
 A. A. Kleshchev, “Against Phase Velocities of Elastic Waves in Thin Transversely Isotropic Cylindrical Shell,” Proceedings of the X session of the Russian Acoustical Society. M.: Geos, Vol. 1, 2000, pp. 206210.
 A. A. Kleshchev, “Diffraction and propagation of waves in Elastic Mediums and Bodies,” Vlas, S.Pb., 2002, p. 156.
 A. A. Kleshchev, “Diffraction, Radiation and Propagation of Elastic Waves,” Profprint, S.Pb., 2006, p. 160.
 E. L. Shenderov, “Radiation and Scattering of Sound,” L.: Shipbuilding, 1989, p. 302.