 | | 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. A. Bazarenko, "Operator Method for Solving the Plane Elasticity Problem in a Domain Bounded by Second-Order Curves," Mech. Solids. 38 (5), 40-49 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
5 |
Pages |
40-49 |
| Title |
Operator Method for Solving the Plane Elasticity Problem in a Domain Bounded by Second-Order Curves |
| Author(s) |
N. A. Bazarenko (Moscow) |
| Abstract |
We consider the conformal mapping z=csinhζ (ζ=u+iv) of a rectangle D* onto the domain D0 bounded by symmetric parts of an ellipse and a hyperbola. In this domain with the elliptic coordinates (u, v), we solve the plane elasticity problem. The biharmonic function F is sought in the form F=F0+xF1+yF2, where Fk=C(v)ψk(u)+S(v)ψ*k(u) are unknown harmonic functions, C(v) and S(v) are operator-valued functions described in [1] and defined by the relation (C(v)+i S(v))ψ(u)=ψ(u+iv). According to the operator method, the unknown functions ψ1 and ψ2 are found from a system of two operator equations, and ψ0 and ψ0* are solutions of first-order differential equations. An arbitrariness involved in ψ*0 and ψk allows one to construct F satisfying all boundary conditions on the boundary of the domain D0. A similar problem was considered in [2], but the solution found there corresponded to the special case of the plate loaded so that the hyperbolic part of the boundary is stress free and the conditions on the elliptic part of the boundary take into account only the resultant forces and torques. |
| References |
| 1. | N. A. Bazarenko, "An operator method for solving
the plane problem of the theory of elasticity,"
Izv. AN. MTT [Mechanics of Solids], No. 3, pp. 73-83, 2000. |
| 2. | H. Neuber, Kerbspannungslehre,
Springer-Verlag, Berlin, 1958. |
| 3. | I. I. Vorovich and O. S. Malkina,
"An asymptotic method for solving an elasticity problem for a thin plate", in
Proc. All-Union Conference on Plates and Shells [in Russian], pp. 251-254, Nauka, Baku, 1966. |
| 4. | V. A. Agarev,
The Method of Initial Functions for Two-dimensional
Boundary Value Problems of Elasticity [in Russian],
Izd-vo AN UkrSSR, Kiev, 1963. |
| 5. | L. V. Kantorovich and V. I. Krylov,
Approximate Methods of Higher Analysis [in Russian], Fizmatgiz, Moscow, Leningrad, 1962. |
|
| Received |
29 January 2001 |
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