 | | 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 57, Issue 3 / 2022 | Next article >> |
| D.V. Georgievskii and N.S. Stetsenko, "Complex Aleksandrovich Representation of Solutions in Displacements in the Three-Dimensional Theory of Elasticity," Mech. Solids. 57 (3), 455-461 (2022) |
| Year |
2022 |
Volume |
57 |
Number |
3 |
Pages |
455-461 |
| DOI |
10.3103/S0025654422030128 |
| Title |
Complex Aleksandrovich Representation of Solutions in Displacements in the Three-Dimensional Theory of Elasticity |
| Author(s) |
D.V. Georgievskii (Lomonosov Moscow State University, Moscow, 119991 Russia; Ishlinsky Institute for Problems in Mechanics RAS, Moscow, 119526 Russia; Moscow Center of Fundamental and Applied Mathematics, Moscow, 119991 Russia, georgiev@mech.math.msu.su)
N.S. Stetsenko (Lomonosov Moscow State University, Moscow, 119991 Russia; Moscow Center of Fundamental and Applied Mathematics, Moscow, 119991 Russia, stetsenkonina@mail.ru) |
| Abstract |
The analytical possibilities of the representation of the solution in displacements in the three-dimensional theory of elasticity in the form of a two-dimensional complex structure, proposed in the 70s of the XX century in the works of A.I. Aleksandrovich, are discussed. Complex-valued displacements are sought in the form of a holomorphic expansion as series in powers of complex variables with antiholomorphic coefficients and in powers of conjugate complex variables with holomorphic coefficients. All holomorphic and antiholomorphic functions are expressed in terms of four arbitrary holomorphic functions. As test special cases leading to classical solutions known in the theory of elasticity, a plane strain state, antiplanar strain, a three-dimensional strain state in a thin plate of variable thickness, axisymmetric displacement fields, which are realized, in particular, under a linear combination of internal (external) pressure, (rθ)-torsion and axial (rz)-shear in a cylindrical layer and (θz)-torsion of a solid cylinder are considered. In terms of complex-valued displacements, a system of equations of the axisymmetric theory of elasticity is written, the fundamental solution of which is a general representation of the displacement field in the axisymmetric case, similarly to the Kolosov-Muskhelishvili formulas in the plane problem. |
| Keywords |
three-dimensional elasticity theory, quasi-statics, Alexandrovich representation, complex displacement, harmonic function, analytic function, axisymmetric field |
| Received |
13 May 2021 | Revised |
18 May 2021 | Accepted |
24 May 2021 |
| Link to Fulltext |
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