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<p>Bodies with cylindrical and plane-parallel boundaries in the form of disks or rings are considered. The general solution of the Navier equations in cylindrical coordinates is used to describe the three-dimensional stress state. The normal and tangential forces are expressed in terms of one two-dimensional harmonic and two biharmonic functions after integrating the stresses over the thickness of the cylindrical plate. The boundary conditions on its plane surfaces and the equilibrium equations are satisfied. It is shown that their averaged stress-strain state is divided into two cases: an axisymmetric state and a&#xa0;non-axisymmetric state, which depend on the polar angle&#xa0;φ. A&#xa0;closed system of partial differential equations for the introduced two-dimensional functions is plotted 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. Analytical formulas for expressing displacements and stresses in these cases are given.</p>

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

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • V. P. Revenko,
  • A. V. Revenko

摘要

Bodies with cylindrical and plane-parallel boundaries in the form of disks or rings are considered. The general solution of the Navier equations in cylindrical coordinates is used to describe the three-dimensional stress state. The normal and tangential forces are expressed in terms of one two-dimensional harmonic and two biharmonic functions after integrating the stresses over the thickness of the cylindrical plate. The boundary conditions on its plane surfaces and the equilibrium equations are satisfied. It is shown that their averaged stress-strain state is divided into two cases: an axisymmetric state and a non-axisymmetric state, which depend on the polar angle φ. A closed system of partial differential equations for the introduced two-dimensional functions is plotted 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. Analytical formulas for expressing displacements and stresses in these cases are given.