A boundary state method for solving a mixed problem of the theory of anisotropic elasticity with mass forces

The paper presents a methodology for determining a stress-strain state of transversely isotropic bodies of revolution under conditions of a mixed problem of the elasticity theory, i.e. displacements of the boundary points are specified on the one part of the surface, and forces are assigned on the other part. At the same time, the body is exposed to mass forces. The problem solving involves the development of the boundary state method. A theory is created to construct the bases of spaces for internal and boundary states. The basis of the internal states includes displacements, strains, and stresses. The basis of the boundary states includes forces at the boundary, displacements of the boundary points, and mass forces. Spaces are conjugated by an isomorphism. It allows one to reduce the determination of the internal state to a study of the boundary state. Characteristics of the stress-strain state are presented in terms of the Fourier series. Finally, the determination of the elastic state is reduced to the solving of an infinite system of algebraic equations.
A result of the study is presented as a solution to the main mixed problem for a hemisphere clamped on a plane surface and exposed to a concentrated compressive force and mass forces.