Solution of the Mixed Problem of the Theory of Elasticity for Anisotropic Bodies with Allowance for Mass Forces

Abstract

The paper presents a technique for determining the stress-strain state of transversely isotropic bodies of revolution under the conditions of a mixed problem of the theory of elasticity, when the displacements of the boundary points are given on one part of the surface, and the forces on the other. At the same time, mass forces act on the area of the body. The fulfillment of this task presupposes the development of the boundary state method. A theory has been developed for constructing the basis of spaces of internal states, including displacements, deformations, stresses and the basis of boundary states, including efforts at the boundary, displacements of points of the boundary and mass forces. The bases are formed on the basis of the general solution of the boundary value problem for a transversely isotropic body of revolution and the method of creating basis vectors of displacement in the problem of determining the state from continuous non-conservative mass forces. The isomorphism of the spaces of internal and boundary states is proved, which makes it possible to unambiguously establish a correspondence between the elements of these spaces. Isomorphism of spaces allows us to reduce the search for an internal state to the study of a boundary state isomorphic to it. The characteristics of the stress-strain state are presented in the form of Fourier series. Fourier coefficients are scalar products that have an energetic meaning: in the space of boundary states, this is the sum of the work of external and mass forces; in the space of internal states, this is the internal energy of elastic deformation. Finally, the search for an elastic state is reduced to solving an infinite system of algebraic equations. The solution of the main mixed problem for a hemisphere clamped on a flat surface and under the action of a compressive force of a local nature and mass forces is presented.