Solution of the Lam\'e problem for combined transversely isotropic spheres with a~general center

A. V. Zaitsev ; Yu. V. Sokolkin ; A. A. Fukalov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 25 (2021) no. 1, pp. 83-96.

Résumé

The paper deals with obtaining an exact analytical solution of the Lamé problem on the equilibrium state of a combined body consisting of two tightly fitted transversely isotropic spheres with a common center. The body is influenced by uniformly distributed external and internal pressures. The process pressure on the contact surface is determined assuming that it is a consequence of the difference in the geometry of the individual parts of the combined sphere only. We analyzed the laws of the influence of the materials’ anisotropy (the material constants satisfy the relations in the form of inequalities that ensure the positivity of the eigenvalues of the elasticity operator) and the values of the contact process pressure on the stress distribution in the cross sections of pressure vessels. The influence assessment of the materials’ anisotropy shows an opportunity to control the values and nature of the stress distribution in the combined structures that are optimal for the specified operating conditions. The obtained results indicate that a change in the anisotropy index, i.e. an increase in its values in the inner or outer parts of the spheres leads to an increase or decrease in the absolute values of stresses, respectively. This increase or decrease in the anisotropy indices can be realized at the stage of structures’ design due to a change in the reinforcement scheme while maintaining the properties of the individual structural elements. Based on a multicriteria approach, the initial strength of combined pressure vessels was estimated using the mechanisms of tension or compression in the radial and hoop directions. It was found that an increase in the pressure on the contact surface can lead to the material domains that do not resist compression in the hoop direction. These domains are located in the vicinity of the internal surface of the vessel, on which a uniformly distributed pressure acts, which is less in the absolute value than the external pressure. It was found that the points of the combined vessel located on the contact surface become most dangerous from the point of beginning the damage by the compression in the radial direction.

Détail

Keywords: analytical solution, Lamé problem, combined transversely isotropic sphere, anisotropy, multi-criteria evaluation of initial strength, combined spherical vessels, contact pressure.

Bibliographie

[1] Lekhnitskii S. G., Theory of Elasticity of an Anisotropic Body, Mir Publ., Moscow, 1981, 430 pp. | MR | Zbl

[2] Saint-Venant B., “Mémoire sur les divers genres d'homogénéité semi-polaire ou cylindrique et sur les homogénéités polaires ou sphéri-coniques et sphériques”, J. Math. Pures Appl., 10 (1865), 297–349

[3] Sharmazanashvili A. Kh., “Calculation of anisotropic thick-walled spherical shells”, Vestn. Inzh. Tekhnikov, 1938, no. 7, 35–37 (In Russian)

[4] Kolchin G. B., Kovalov E. K., “Centrally symmetric deformation of an elastic radially inhomogeneous transversely isotropic hollow sphere”, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1995, no. 6, 42–47 (In Russian)

[5] Zaitsev A. V., Fukalov A. A., “Elastic equilibrium state of thick-walled heavy transversely-isotropic spheres fixed on the interior surface”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2010, no. 5(21), 85–95 (In Russian) | DOI | Zbl

[6] Zaytsev A. V., Sokolkin Y. V., Fukalov A. A., “Initial damage mechanisms of reinforced concrete monolithic supports for spherical mine workings located in sedimentary rock mass”, PNRPU Mechanics Bulletin, 2013, no. 4, 54–74 (In Russian) | DOI

[7] Zaitsev A. V., Fukalov A. A., Sokolkin Y. V., “Initial strength analysis of anisotropic concrete supports for spherical mine workings in a sedimentary rock mass”, Physical and Mathematical Modeling of Earth and Environment Processes, Springer, Cham, 2019, 463–471 | DOI

[8] Pobedrya B. E., Mekhanika kompozitsionnykh materialov [Mechanics of Composite Materials], Moscow State Univ., Moscow, 1984, 336 pp. (In Russian)

[9] Vildeman V. E., Sokolkin Yu. V., Tashkinov A. A., Mekhanika neuprugogo deformirovaniia i razrusheniia kompozitsionnykh materialov [Mechanics of Inelastic Deformation and Fracture of Composite Materials], Nauka, Moscow, 1997, 288 pp. (In Russian)

[10] Mityushov E. A., Berestova S. A., Odintsova N. Yu., “Effective elastic properties of textured cubic polycrystals”, Texture, Stress, and Microstructure, 35:2 (2002), 99–111 | DOI

[11] Bobritskii N. V., Yufin V. A., Osnovy neftianoi i gazovoi promyshlennosti [Fundamentals of Oil and Gas Industry], Nedra, Moscow, 1988, 200 pp. (In Russian)