Calculation of the stress state of cylindrical bodies based on the three-dimensional theory of elasticity

Keywords

disk, ring, plate, averaged stress state, polar coordinate system, force, boundary conditions.

Cite as

Revenko V. P. and Revenko А. V. Calculation of the stress state of cylindrical bodies based on the three-dimensional theory of elasticity. Physicochemical Mechanics of Materials. 2025. 61(5), 109-115.

https://doi.org/10.15407/pcmm2025.05.109

Abstract

Bodies with cylindrical and plane-parallel boundaries in the form of disks or rings are considered. To describe their three-dimensional stress state, the general solution of the Navier equations in the cylindrical coordinate system is used. After integrating the stresses over the thickness of the cylindrical plate, the normal and tangential forces are expressed in terms of one two-dimensional harmonic and two biharmonic functions. The boundary conditions on its plane surfaces and the equilibrium equations are satisfied. A closed system of partial differential equations for the introduced two-dimensional functions is constructed without using hypotheses about the geometric nature of the plate deformation. An averaged two-dimensional representation of the general solution in the polar coordi­nate system based on the three-dimensional theory of elasticity is obtained for the first time. New solutions of the equilibrium equations of plates in the polar coordinate system are constructed. It is shown that their averaged stress-strain state is divided into two cases: an axisymmetric stress state, and a non-axisymmetric state, which depends on the polar angle φ. Analytical formulas are given for expressing displacements and stresses in these cases.

References

1. S. P. Timoshenko, and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York (1970).
2. M. H. Sadd, Elasticity: Theory, Applications, and Numeric, Acad. Press, Burlington (2009).
3. A. S. Kosmodamianskyi, and V. A. Shangyrvan, Thick Multi-Connected Plates [in Russian], Naukiva Dumka, Kyiv (1978).
4. S. P. Timoshenko, and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York (1959).
5. L. Y. Donnell, Beams, Plates and Shells, McGraw-Hill Inc., Chicago (1976).
6. V. V. Pabasyuk, M. M. Stadnyk, and V. P. Sylovanuyk, Stress Concentrations in Tree-Dimensional Bodies with Thin Inclusions [in Russain], Naukova Dumka, Kyiv (1986).
7. H. A. Kobayashi, “Survey of Books and Monographs on Plates,” Mem. Fac. Eng., 38, 73-98 (1997). https://doi.org/10.1080/04597239708461060
8. Lukasiewicz S. Local Loads in Plates and Shells. Monographs and Textbooks on Mechanics of Solids and Fluids, Sijthoff & Noordhoff, Alphen aan den Rijn (1979).
9. H. Chen, and L. X. Cai, “Unified ring-compression model for determining tensile properties of tubular materials,” Mater. Today Communications, 13, 210-220 (2017). https://doi.org/10.1016/j.mtcomm.2017.10.006
10. W. Wang, and M. X. Shi, “Thick plate theory based on general solutions of elasticity,” Acta Mechanica, 123, 27-36 (1997). https://doi.org/10.1007/BF01178398
11. V. P. Revenko and A. V. Revenko, “Determination of plane stress-strain states of the plates on the basis of the three-dimensional theory of elasticity,” Mater. Sci., 52, Is. 6, 811-818 (2017). https://doi.org/10.1007/s11003-017-0025-7
12. V. P. Revenko, “Solving the three-dimensional equations of the linear theory of elasticity,” Int. Appl. Mech., 45, Is. 7, 730-741 (2009). https://doi.org/10.1007/s10778-009-0225-4
13. V. Revenko, “Finding physically justified partial solutions of the equations of the thermoelasticity theory in the cylindrical coordinate system,” Scientific J. of Ternopil National Technical University, 112, Is. 4, 58-66 (2023). https://doi.org/10.33108/visnyk_tntu2023.04.058