 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| M.Sh. Israilov, "Reduction of Boundary Value Problems of Dynamic Elasticity to Scalar Problems for Wave Potentials in Curvilinear Coordinates," Mech. Solids. 46 (1), 104-108 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
1 |
Pages |
104-108 |
| DOI |
10.3103/S002565441101016X |
| Title |
Reduction of Boundary Value Problems of Dynamic Elasticity to Scalar Problems for Wave Potentials in Curvilinear Coordinates |
| Author(s) |
M.Sh. Israilov (Academician Millionschikov Grozny State Oil Institute, A. Avtorkhanova 14/53, Groznyy, Chechenskaya resp., 364052 Russia, israiler@hotmail.com) |
| Abstract |
Owing to significant mathematical difficulties arising when solving dynamic problems of elasticity, ever more attention is paid to the study of types of boundary value problems, boundary shapes, and additional assumptions (for example, such as symmetry) for which, in the statement of the problem in potentials, not only the equations of motion lead to separate scalar wave equations but also the boundary conditions split into separate conditions for each of the potentials.
It was shown earlier that the boundary conditions prescribing the normal displacement and tangential stresses (condition (a)) or the normal stress and tangential displacements (condition (b)) on the boundary can be separated for potentials on the plane boundary. In connection with the separation of these boundary conditions on curvilinear boundaries, many claims were made in the literature, some of which are erroneous. In the present paper, we obtain the most complete result, which clarifies this problem and states that the boundary conditions (a) can be separated on the surface of a circular cylinder and a circular cone in the case of axial symmetry, but the boundary conditions (b) cannot be separated on a curvilinear boundary.
Several examples illustrating the obtained results are given. |
| Keywords |
dynamic theory of elasticity, wave potentials, boundary value problems, curvilinear coordinates |
| References |
| 1. | Y.-H. Pao and C.-C. Mow,
Diffraction of Elastic Waves and Dynamic Stress Concentration
(Grane Russak, New York, 1973; Adam Hilger, London, 1973). |
| 2. | L. M. Flitman,
"On a Boundary Value Problem for Elastic Half-Space,"
Izv. Akad. Nauk SSSR. Ser. Geogr. Geofiz.
No. 1, 105-106 (1958). |
| 3. | V. B. Poruchikov,
"Diffraction of Spherical Elastic Wave on a Cone,"
Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela,
No. 2, 154-157 (1976). |
| 4. | V. B. Poruchikov,
Methods of Dynamical Elasticity
(Nauka, Moscow, 1986)
[in Russian]. |
| 5. | A. E. H. Love,
Treatise on the Mathematical Theory of Elasticity, 4th ed.
(Cambridge Univ. Press, Cambridge, 1927; ONTI, Moscow-Leningrad, 1935). |
| 6. | M. Sh. Israilov,
Dynamical Theory of Elasticity and Wave Diffraction
(Izd-vo MGU, Moscow, 1992)
[in Russian]. |
| 7. | D. S. Jones,
The Theory of Electromagnetism
(Pergamon Press, Oxford, 1964). |
| 8. | E. W. Hobson,
The Theory of Spherical and Ellipsoidal Harmonics
(Cambridge Univ. Press, Cambridge, 1931; IL, Moscow, 1952). |
| 9. | H. M. Macdonald,
"Zeros of the Spherical Harmonic Pnrn(μ)
Considered as a Function of n,"
Proc. Lond. Math. Soc.
V. XXXI (1), 264-278 (1900). |
|
| Received |
15 August 2010 |
| Link to Fulltext |
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