DOI: https://doi.org/10.32347/2410-2547.2019.102.77-90

### The study of the second main problem of the theory of elasticity for a layer with a cylindrical cavity

Vitalii Miroshnikov

#### Abstract

The stress - strain state of a layer with a cylindrical cavity was investigated, when displacements are set at the boundaries of the layer and at the boundary of the cavity. On the cavity and on the boundaries of the layer, displacements are given.

The solution of the spatial problem of the theory of elasticity is obtained by generalized Fourier method in relation to the system of Lamex equations in cylindrical coordinates connected with the cylinder, and Cartesian coordinates associated with the boundaries of the layer. Infinite systems of linear algebraic equations obtained as a result of satisfaction of boundary conditions, solved by the truncation method. As a result, movements and strains were obtained at different points of the elastic body.

Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the method of reduction, which is proved by the high accuracy of the implementation of boundary conditions.

A numerical analysis of the stress - strain state of the body at various distances from the cylindrical cavity to the boundaries of the layer is carried out.

The greatest normal stresses occur on the surface of the cylindrical cavity and on the isthmus between the surface of the cylindrical cavity and the boundaries of the layer. As the boundary surfaces approach each other, the stresses increase.

The greatest tangential stresses  arise on the surface of the cylindrical cavity at φ = 5π / 16, φ = 11π / 16, φ = 21π / 16 and φ = 27π / 16, and with the approximation of the boundary surfaces to each other the stress  increases.

The greatest tangential stresses  arise on the surface of the cylindrical cavity along the z axis at φ = 0 and at φ = p. When the boundaries of the layer approximation the cylindrical cavity  decreases.

The given analysis and algorithm of calculation can be used in the design of constructions, in the calculation schemes of which there is a layer with a cylindrical cavity and specified on the boundary surfaces by displacements.

The stress - strain state of a layer with a cylindrical cavity was investigated, when displacements are set at the boundaries of the layer and at the boundary of the cavity. On the cavity and on the boundaries of the layer, displacements are given.

The solution of the spatial problem of the theory of elasticity is obtained by generalized Fourier method in relation to the system of Lamex equations in cylindrical coordinates connected with the cylinder, and Cartesian coordinates associated with the boundaries of the layer. Infinite systems of linear algebraic equations obtained as a result of satisfaction of boundary conditions, solved by the truncation method. As a result, movements and strains were obtained at different points of the elastic body.

Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the method of reduction, which is proved by the high accuracy of the implementation of boundary conditions.

A numerical analysis of the stress - strain state of the body at various distances from the cylindrical cavity to the boundaries of the layer is carried out.

The greatest normal stresses occur on the surface of the cylindrical cavity and on the isthmus between the surface of the cylindrical cavity and the boundaries of the layer. As the boundary surfaces approach each other, the stresses increase.

The greatest tangential stresses  arise on the surface of the cylindrical cavity at φ = 5π / 16, φ = 11π / 16, φ = 21π / 16 and φ = 27π / 16, and with the approximation of the boundary surfaces to each other the stress  increases.

The greatest tangential stresses  arise on the surface of the cylindrical cavity along the z axis at φ = 0 and at φ = p. When the boundaries of the layer approximation the cylindrical cavity  decreases.

The given analysis and algorithm of calculation can be used in the design of constructions, in the calculation schemes of which there is a layer with a cylindrical cavity and specified on the boundary surfaces by displacements.

#### Keywords

cylindrical cavity in a layer; Lame's equation; generalized Fourier method; infinite systems of linear algebraic equations

PDF (Українська)

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