Solution of a mixed nonaxisymmetric problem of the theory of elasticity for anisotropic bodies of revolution

Abstract

The paper developed a technique for solving mixed nonaxisymmetric problems of the theory of elasticity for bounded bodies of revolution made of a transversely isotropic material under the action of surface forces specified according to a cyclic law. The technique involves the development of the energy method of boundary states, which is based on the concepts of spaces of internal and boundary states, conjugated by isomorphism, which makes it possible to establish a one-to-one correspondence between the elements of these spaces. The internal state includes the components of the tensor of stresses, deformations, and the displacement vector. The boundary state includes efforts and displacements at the boundary of the body. The isomorphism of the state spaces is proved, which allows finding the internal state to be reduced to the study of the boundary state isomorphic to it. The basis is formed on the basis of the general solution of the boundary value problem of elastostatics for a transversely isotropic body of revolution. Orthogonalization of state spaces is carried out, where the internal energy of elastic deformation is used as scalar products in the space of internal states; in the space of boundary states, the work of external forces is used. Finally, finding the desired state is reduced to solving an infinite system of algebraic equations for the Fourier coefficients. The solution of the problem with mixed boundary conditions for a circular in plan cylinder of transversely isotropic coarse dark gray siltstone with anisotropy axis coinciding with the geometric axis of symmetry is presented. The solution is analytical and the characteristics of the stress-strain state have a polynomial form. Explicit and indirect signs of convergence of problem solutions and graphical visualization of the results are presented.