A NEW SOLUTION TO THE PROBLEM OF THE THEORY OF ELASTICITY IN STRESSES ABOUT A ROTATING ROTATION ELLIPSOID

Abstract

Abstract. An exact semi-inverse solution of the fundamental axisymmetric spatial problem of the mechanics of an elastically deformable solid medium on the uniform rotation around a fixed axis of a weightless ellipsoid made of a homogeneous isotropic material obeying the generalized Hooke law, taking into account the mass centrifugal forces of inertia was obtained based on the two-dimensional equilibrium equations and modified conditions of deformations continuity.  The uncertainty (incorrectness) associated with the multivariance of the approximate solution of the problem by the inverse method, depending on the choice of approximating expressions for the stress function of the homogeneous biharmonic Love’s equation and the selection of the corresponding internal force factors for the particular solution has been eliminated. Compact and physically justified stress formulas in the form of four quadratic polynomials satisfying the system of resolving differential equations and all boundary conditions on the ellipsoid surface have been derived. A numerical evaluation of the known theory has been carried out, in comparison with the proposed new solution for three values of the Poisson's ratio, covering the permissible range from 0 to 0.5, for the special case of a rotating ball (when the semiaxis of the ellipse are equal). In the process of this comparison, a sufficiently large error has been established, even from an engineering point of view, of the existing mathematical model, which, according to the extreme main stresses acting in the center of a spherical body, ranges from 16.22% to 22.82%. The great importance and necessity of the practical application of the developed classical mechanical and mathematical model to the high-precision calculation of the strength of the  rapidly rotating turbine disks of elliptical profile in order to increase their guaranteed load-bearing capacity and reliability during the operation  has been noted and illustrated.