A discussion on “representing general solution of equations in theory of elasticity by harmonic functions”

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

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Complex representation of general solution of equations for nonlinear model of plane deformation of crystal media with a complex lattice

2018 Days on Diffraction (DD) ◽

10.1109/dd.2018.8553193 ◽

2018 ◽

Author(s):

Anatolii N. Bulygin ◽

Yurii V. Pavlov

Keyword(s):

General Solution ◽

Nonlinear Model ◽

Plane Deformation ◽

Complex Representation ◽

Complex Lattice ◽

Solution Of Equations

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Spherical Symmetry in the Theory of Elasticity

Journal of Applied Mechanics ◽

10.1115/1.4011702 ◽

1958 ◽

Vol 25 (1) ◽

pp. 136-140

Author(s):

F. Goded

Keyword(s):

General Solution ◽

Stress Tensor ◽

Axial Symmetry ◽

Spherical Symmetry ◽

Stress Function ◽

Theory Of Elasticity ◽

Plane Symmetry ◽

The Subject ◽

The Common

Abstract Part 1 of the paper presents a study of the common characteristics of both plane symmetry and axial symmetry in the theory of elasticity; viz., the form of the stress tensor and the existence of further analogous symmetries. In Part 2, the subject deals with the possibility of the existence of further analogous symmetries which are found to be possible only in some specific cases. In particular, spherical symmetry is treated. The method of obtaining the stress function of this new symmetry and the equation which this function must satisfy also are discussed, together with the stresses expressed by means of this stress function. The paper ends with a brief review of a general solution of the stress function and an example of the application of this stress function to a given problem.

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Separation of the 3D stress state of a loaded plate into two-dimensional tasks: bending and symmetric compression of the plate

Scientific journal of the Ternopil national technical university ◽

10.33108/visnyk_tntu2021.03.053 ◽

2021 ◽

Vol 103 (3) ◽

pp. 53-62

Author(s):

Victor Revenko ◽

Andrian Revenko

Keyword(s):

Boundary Conditions ◽

Harmonic Functions ◽

Three Dimensional ◽

Partial Solution ◽

Theory Of Elasticity ◽

Two Dimensional ◽

Normal Stresses ◽

Equilibrium Equations ◽

Dimensional Boundary ◽

The Right

The three-dimensional stress-strain state of an isotropic plate loaded on all its surfaces is considered in the article. The initial problem is divided into two ones: symmetrical bending of the plate and a symmetrical compression of the plate, by specified loads. It is shown that the plane problem of the theory of elasticity is a special case of the second task. To solve the second task, the symmetry of normal stresses is used. Boundary conditions on plane surfaces are satisfied and harmonic conditions are obtained for some functions. Expressions of effort were found after integrating three-dimensional stresses that satisfy three equilibrium equations. For a thin plate, a closed system of equations was obtained to determine the harmonic functions. Displacements and stresses in the plate were expressed in two two-dimensional harmonic functions and a partial solution of the Laplace equation with the right-hand side, which is determined by the end loads. Three-dimensional boundary conditions were reduced to two-dimensional ones. The formula was found for experimental determination of the sum of normal stresses via the displacements of the surface of the plate.

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