 | | 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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| << Previous article | Volume 40, Issue 4 / 2005 | Next article >> |
| Yu. A. Bogan, "Regular integral equations for the second boundary value problem in two-dimensional anisotropic elasticity," Mech. Solids. 40 (4), 13-19 (2005) |
| Year |
2005 |
Volume |
40 |
Number |
4 |
Pages |
13-19 |
| Title |
Regular integral equations for the second boundary value problem in two-dimensional anisotropic elasticity |
| Author(s) |
Yu. A. Bogan (Novosibirsk) |
| Abstract |
Regular integral equations (Fredholm equations of the second kind) are
constructed for the solution of the second boundary value problem of
anisotropic elasticity (the displacement vector is specified on the
boundary) in a simply connected bounded domain with Lyapunov boundary on
the plane. Note that similar equations for an isotropic medium were
constructed by a different method in [1]. It turns out that a complete
potential theory similar to that for one second-order elliptic equation can
be constructed under minimal smoothness assumptions for the boundary and
the boundary data in the second boundary value problem (if the problem is
to be solved in a Hölder function class). Recall the essentials. Let
Di be the solution of the interior Dirichlet problem for the Laplace
equation, and let Ne be the solution of the exterior Neumann problem.
The integral equations corresponding to these problems form a Fredholm
adjoint pair (e.g., see [3]). It is such a pair of equations that is
constructed in the present paper for the second boundary value problem of
elasticity. As is shown below, the medium anisotropy simplifies the formal
algebraic aspects dramatically. Part of the equations given here were
earlier published in [4], but the analysis carried out there is far less
complete than in the present paper. Note that an attempt to construct
similar equations was made in [5], but the paper lacks sound proofs. For
example, the Fredholm property of the equations constructed in [5] is not
proved there; nor is the equivalence of these equations to the original
boundary value problem shown. |
| References |
| 1. | Ya. B. Lopatinskii, "On a method for solving the second
fundamental problem of the theory of elasticity," Teoret. Prikl. Mat.,
No. 1, pp. 23-27, 1958. |
| 2. | D. I. Sherman, "On the solution of the static plane elasticity
problem with prescribed displacements on the boundary," Doklady AN SSSR,
Vol. 27, No. 9, pp. 911-913, 1940. |
| 3. | I. G. Petrovskii, Lectures on Partial Differential Equations [in
Russian], Fizmatgiz, Moscow, 1961. |
| 4. | Yu. A. Bogan, "On Fredholm integral equations in two-dimensional
anisotropic elasticity," Sib. Zh. Vychisl. Matem., Vol. 4, No. 1,
pp. 21-30, 2001. |
| 5. | M. O. Basheleishvili, "Solution of plane boundary value problems
of the statics of an anisotropic elastic body," Trudy VTs AN GSSR, Vol. 3,
pp. 93-39, 1963. |
| 6. | S. G. Lekhnitskii, Anisotropic Plates [in Russian], OGIZ,
Moscow, Leningrad, 1947. |
| 7. | N. I. Muskhelishvii, Singuar Integral Equations [in Russian],
Nauka, Moscow, 1968. |
| 8. | I. N. Vekua, Generalized Analytic Functions [in Russian],
Fizmatgiz, Moscow, 1959. |
|
| Received |
04 July 2003 |
| << Previous article | Volume 40, Issue 4 / 2005 | Next article >> |
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