 | | 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
|
| In English (Mech. Solids): | | 4760 |
|
| << Previous article | Volume 53, Issue 4 / 2018 | Next article >> |
| V.V. Vasiliev, "Singular Solutions in the Problems of Mechanics and Mathematical Physics," Mech. Solids. 53 (4), 397-410 (2018) |
| Year |
2018 |
Volume |
53 |
Number |
4 |
Pages |
397-410 |
| DOI |
10.3103/S0025654418040052 |
| Title |
Singular Solutions in the Problems of Mechanics and Mathematical Physics |
| Author(s) |
V.V. Vasiliev (Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia, vvvas@dol.ru) |
| Abstract |
A problem of the solutions singularity for applied problems is discussed. It is proposed to qualify such solutions as formal mathematical results that arise from the discrepancy between the mathematical and physical models of the phenomenon or object being studied. As examples, we consider the singular solution of the Schwarzschild problem in the general theory of relativity (serving as the mathematical basis for the existence of objects called the Black Holes), the solution of the mathematical physics problem for a circular membrane loaded in the center by a concentrated force, and the solution for the problems of the theory of elasticity about a cylindrical punch and an expandable plate with a crack. A generalization of the classical definition for a function and its derivative is proposed. This generalization makes it possible to obtain regular solutions of traditional singular problems. |
| Keywords |
general theory of relativity, mathematical physics, theory of elasticity, singular solutions |
| References |
| 1. | I.D. Novikov,
Black Holes and the Universe
(Molodaya Gvardia, Moscow, 1985) [in Russian]. |
| 2. | K. Thorne,
Black Holes and Time Warps
(Fizmatgiz, Moscow, 2009) [in Russian]. |
| 3. | S.L. Sobolev,
Equations of Mathematical Physics
(GITTL, Moscow, Leningrad, 1950) [in Russian]. |
| 4. | D. Singh,
General Relativity
(Izd-vo Inostr. Lit-ry, Moscow, 1963) [in Russian]. |
| 5. | A. Einstein,
"On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses,"
Ann. Math.
40 (4), 922-936 (1939). |
| 6. | http://singularnost.net |
| 7. | V.V. Vasiliev and L.V. Fedorov,
"Stress State of an Elastic Ball in a Spherically Symmetric Gravitational Field,"
Izv. Akad. Nauk. Mekh. Tverd. Tela
No. 4, 15-29 (2014)
[Mech. Sol. (Engl. Transl.) 49 (4), 379-381 (2014)]. |
| 8. | S.G. Mikhlin,
Linear Partial Differential Equations
(Vyshshaya Shkola, Moscow, 1977) [in Russian]. |
| 9. | Theory of Flexible Round plates
(IL, Moscow, 1967) [in Russian]. |
| 10. | V.V. Vasiliev and L.V. Fedorov,
"Solution of a Spherically Symmetric Static Problem of General Relativity for an Elastic Solid Sphere,"
Appl. Phys. Res.
9 (6), 8-13 (2017). |
| 11. | G.P. Cherepanov,
"Singular Solutions in the Theory of Elasticity,"
in Mechanics of Solids
(Sudostroenie, Leningrad, 1970), pp. 467-479 [in Russian]. |
| | |
| 12. | A. Love,
Mathematical Theory of Elasticity
(ONTTI, Moscow, Leningrad, 1935) [in Russian]. |
| 13. | A. Feppl and L. Feppl,
Strength and Deformation. Applied Theory Elasticity
(GTTI, Moscow, 1933) [in Russian]. |
| 14. | A.I. Lurie,
Spatial Problems of Elasticity Theory
(GITTL, Moscow, 1955) [in Russian]. |
| 15. | S.P. Timoshenko and J.N. Goodier,
Theory of Elasticity
(Nauka, Moscow, 1975) [in Russian]. |
| 16. | Yu.N. Rabotnov,
Mechanics of a Deformable Solid
(Nauka, Moscow, 1979) [in Russian]. |
| 17. | Fracture. Edited by E. Libowitz. Vol. 2. Mathematical Fundamentals
(AcademicPress, New York and London, 1968). |
| 18. | S.A. Lurie and V.V. Vasiliev,
The Biharmonic Problem in the Theory of Elasticity
(Gordon and Breach Publishers, 1995). |
| 19. | V.V. Vasiliev and S.A. Lurie,
"Continuous Model with Microstructure,"
Komp. Nanost.
7 (1), 25-33 (2015). |
| 20. | V.V. Vasiliev and S.A. Lurie,
"Generalized Theory of Elasticity,"
Izv. Akad. Nauk. Mekh. Tverd. Tela
No. 4, 16-27 (2015)
[Mech. Solids (Engl. Transl.) 50 (4), 379-388 (2015)] |
| 21. | V.V. Vasiliev and S.A. Lurie,
"New Solution of Axisymmetric Contact Problem of Elasticity,"
Izv. Akad. Nauk. Mekh. Tverd. Tela
No. 5, 12-21 (2017)
[Mech. Solids (Engl. Transl.) 52 (5), 479-487 (2017)] |
| 22. | A.V. Andreev,
Engineering Methods for Determining Stress Concentration in Machine Elements
(Mashinostroenie, Moscow, 1976) [in Russian]. |
| 23. | V.V. Vasiliev and S.A. Lurie,
"Generalized Solution of the Problem on a Circular Membrane Loaded by a Concentrated Force,"
Izv. Akad. Nauk. Mekh. Tverd. Tela
No. 3, 115-119 (2016)
[Mech. Solids (Engl. Transl.) 51 (3), 334-338 (2016)]. |
| 24. | V.V. Vasiliev and S.A. Lurie
"New Solution of the Plane Problem for an Equilibrium Crack,"
Izv. Akad. Nauk. Mekh. Tverd. Tela
No. 5, 61-67 (2016)
[Mech. Solids (Engl. Transl.) 51 (5), 557-561 (2016)]. |
| 25. | G.P. Cherepanov,
Mechanics of Brittle Fracture
(Nauka, Moscow, 1974) [in Russian]. |
|
| Received |
05 March 2018 |
| Link to Fulltext |
|
| << Previous article | Volume 53, Issue 4 / 2018 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English