Solution of the Second Main Axisymmetric Problem of Elasticity Theory for Anisotropic Bodies of Rotation

The paper presents a mathematical model for solving the second main problem of the theory of elasticity for limited bodies of revolution made of a transversally isotropic material. Non-axisymmetric kinematic conditions are imposed on the surface of the body, specified according to the cyclic law of sine and cosine. 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, coupled by isomorphism. Isomorphism of state spaces allows us to establish the one-to-one correspondence between the elements of these spaces. The internal state includes the components of the stress tensor, strain tensor and displacement vector. The boundary state includes surface forces and displacements of the body boundary points. Finding an internal state comes down to studying the boundary state isomorphic to it. The basis of the internal states is reduced on the basis of a general solution to the boundary value problem of elastostatics for a transversely isotropic body limited by coaxial surfaces 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 surface forces on the displacements of points on the boundary of the body is used. Finally, finding the desired state comes down to solving an infinite system of linear algebraic equations with respect to the Fourier coefficients. A solution to the second main problem is presented with boundary conditions simulating transverse expansion (without longitudinal compression) for a circular cylinder made of transversely isotropic large dark gray siltstone. The solution is analytical; the characteristics of the stress-strain state have a polynomial form. Explicit and indirect signs of convergence of the problem solution and graphical visualization of the results are presented.