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IssuesArchive of Issues2022-3pp.455-461

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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 2021Revised 18 May 2021Accepted 24 May 2021
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