 | | 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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| N.I. Martynov, "Integral Equations of Plane Static Boundary Value Problems of the Elasticity Theory for an Inhomogeneous Anisotropic Medium," Mech. Solids. 51 (4), 451-471 (2016) |
| Year |
2016 |
Volume |
51 |
Number |
4 |
Pages |
451-471 |
| DOI |
10.3103/S0025654416040087 |
| Title |
Integral Equations of Plane Static Boundary Value Problems of the Elasticity Theory for an Inhomogeneous Anisotropic Medium |
| Author(s) |
N.I. Martynov (Institute of Mathematics and Mathematical Modeling, ul. Pushkina 125, Almaty, 050010 Kazakhstan, nirmar50@mail.ru) |
| Abstract |
The static boundary value problems of plane elasticity for an inhomogeneous anisotropic medium in a simply connected domain are reduced to the Riemann-Hilbert problem for a quasi-analytic vector. Singular integral equations over the domain are obtained, and their solvability is proved for a sufficiently wide anisotropy class. In the case of a homogeneous anisotropic body, the solutions of the first and second boundary value problems are obtained in closed form.
For compound elastic media with anisotropy varying over a domain (of a sufficiently wide class), uniquely solvable integral equations of boundary value problems of static elasticity for an inhomogeneous anisotropic medium are obtained, which readily permits finding generalized solutions that satisfy the matching conditions on the interfaces between the subdomains. |
| Keywords |
anisotropic body, integral equation, boundary value problem, index, Riemann-Hilbert problem |
| References |
| 1. | L. Bers,
"Partial Differential Equations and Generalized Analytic Functions,"
Proc. Nat. Ac. Se. USA
37 (1), 42-47 (1951). |
| 2. | I. N. Vekua,
Generalized Analytic Functions
(Nauka, Moscow, 1988)
[in Russian]. |
| 3. | B. B. Bojarski,
"Theory of Generalized Analytic Vector,"
Annales Polonici Mathematicy
17, 281-320 (1966). |
| 4. | N. I. Monakhov, Boundary Value Problems with Free
Boundaries for Elliptic Systems of Equations (Nauka, Moscow, 1977)
[in Russian]. |
| 5. | N. I. Monakhov,
"Nonlinear Diffusion Processes,"
Sibirsk. Mat. Zh.
44 (5), 1082-1097 (2003)
[Siberian Math. J. (Engl. Transl.)
44 (5), 845-856 (2003)]. |
| 6. | N. I. Muskhelishvili,
Some Fundamental Problems of Mathematical Elasticity Theory
(Nauka, Moscow, 1966)
[in Russian]. |
| 7. | N. I. Martynov, "Boundary Value Problems of Elasticity of
Inhomogeneous Medium Treated as Boundary Value Problems of
Generalized Analytic Vector," Mat. Zh., No. 3(25), 69-77 (2007). |
| 8. | N. I. Martynov, "Reduction of Boundary Value Problems of
Elasticity to boundary Value Problems of Generalized Analytic
Vector," in Theses of Intern. Sci. Conf. "Differential Equations, Theory of Functions, and Applications Dedicated to Academician I. N. Vekua on the Occasion of His 100th Birthday
(2007), pp. 518-519 [in Russian]. |
| 9. | L. A. Alekseeva, N. I. Martynov, and I. Yu. Fedorov,
"Application of Quasiconformal Mapping in Problems of Torsion
of Inhomogeneous Anisotropic Bodies,"
Mat. Zh.
9 (3(33)), 14-18 (2009). |
| 10. | K. F. Chernykh,
Nonlinear Elasticity in Engineering
(Mashinostroenie, Leningrad, 1986)
[in Russian]. |
| 11. | S. G. Lekhnitskii,
Theory of Elasticity of Anisotropic Bodies
(Nauka, Moscow, 1977)
[in Russian]. |
| 12. | K. F. Chernykh,
Introduction to Anisotropic Elasticity
(Nauka, Moscow, 1988)
[in Russian]. |
| 13. | N. I. Ostrosablin,
"Canonical Moduli and General Solution of Equations of a Two-Dimensional Static Problem of
Anisotropic Elasticity,"
Zh. Prikl. Mekh. Tekhn. Fiz.
51 (3), 94-106 (2010)
[J. Appl. Mech. Tech. Phys. (Engl. Transl.)
51 (3), 377-388 (2010)]. |
| 14. | I. G. Petrovskii,
Lectures on the Theory of Partial Differential Equations
(Gos. Tekh. Izd. Tekh.-Teor. Lit., Moscow, 1953)
[in Russian]. |
| 15. | N. I. Muskhelishvili, Singular Integral Equations (Boundary Value Problems of Theory of Functions and Some of Their Applications in Mathematical Physics (Fiz.-Mat. Lit., Moscow,
1962) [in Russian]. |
| 16. | N. P. Vekua, Systems of Singular Integral Equations and Some Boundary Value Problems (Nauka, Moscow, 1970) [in Russian]. |
| 17. | F. D. Gakhov, Boundary Value Problems (Nauka, Moscow,
1977) [in Russian]. |
| 18. | S. N. Antontsev and N. I. Monakhov, "Boundary Value Problems with
Discontinuous Boundary Conditions for Quasilinear Elliptic Systems
of 2m (m≥1) First-Order Equations," Izv. SO AN SSSR. Ser.
Tekhn. Nauk 8 (2), 65-73 (1967). |
| 19. | Ch. Ashyraliev and N. I. Monakhov,
"Iteration Algorithm for Solving Two-Dimensional Singular Integral Equations,"
Din. Sploshnoi Sredy,
No. 101, 21-29 (1991). |
| 20. | G. N. Savin,
Stress Distribution near Holes
(Naukova Dumka, Kiev, 1968)
[in Russian]. |
| 21. | G. S. Litvinchuk, Boundary Value Problems and Singular Integral Equations with Translation (Nauka, Moscow, 1977) [in Russian]. |
| 22. | E. A. Raenko, "Boundary Value Problems for a Quasiholomorphic
Vector," Din. Sploshnoi Sredy, No. 118, 65-68 (2001). |
| 23. | O. A. Ladyzhenskaya and N. I. Ural'tseva,
Linear and Quasilinear Equations of Elliptic Type
(Nauka, Moscow, 1964)
[in Russian]. |
| 24. | V. A. Lomakin,
Theory of Elasticity of Inhomogeneous Bodies
(Izdat. MGU, Moscow, 1976)
[in Russian]. |
| 25. | N. I. Martynov,
"Quasiconformal Mappings in Plane Elasticity of Inhomogeneous Anisotropic Medium,"
Vestnik NAN RK,
No. 5, 11-19 (2012). |
|
| Received |
18 April 2012 |
| Link to Fulltext |
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