Solution of the Nonaxisymmetric Elastostatic Problem for a Transversely Isotropic Body of Revolution

Abstract

The paper investigates the elastic equilibrium of transversely isotropic bodies of revolution under the action of stationary surface forces distributed according to the cyclic law. The proposed method for constructing the stress-strain state is a development of the method of boundary states. The method is based on the concept of spaces of internal and boundary states conjugated by an isomorphism. Bases of state spaces are formed and orthonormalized. The desired state is expanded in a series by the elements of the orthonormal basis, and the Fourier coefficients, which are quadratures, of this linear combination are calculated. The basis of the internal state space relies on the general solution of the problem of plane deformation of a transversely isotropic body and the formulas for the transition to a spatial state, the components of which depend on three coordinates. Scalar products in state spaces represent the internal energy of elastic deformation and the work of surface forces on the displacements of the boundary points. The study introduces the solution of the main mixed problem for a circular cylinder made of transversely isotropic siltstone with the axis of anisotropy coinciding with the geometric axis of symmetry. The solution is analytical and the characteristics of the stress-strain state have a polynomial form. The paper presents explicit and indirect signs of convergence of problem solutions and graphically visualizes the results