On the stability of equilibrium in a compressed sphere
Using the general three-dimensional formulation of finite deformation theory, an analytical algorithm for transforming a linearized equilibrium stability system in a compressed sphere is proposed for an arbitrary viscoelastic potential. The stress and strain state of the medium is described using the asymmetric Piola-Kirchhoff stress tensor and the Green strain tensor components. Study of the stability of equilibrium is divided into the basic (unperturbed) state and the perturbed stress and strain state, which is the stable form of equilibrium and is described by relevant linearized relations. The solution of the problem in the spherical coordinate system is obtained using the method of separation of variables with respect to the radial displacement, displacement due to rotation, the resulting strain in the principal directions and to their velocities. Equilibrium stability is determined dynamically. Numerical and graphical analysis for stress and strain state is carried out for a compressed elastic three-layer sphere. The analysis is used to investigate the tectonic effects of its own gravitation and the internal follow-up pressure at boundaries of the spheres, which model the lithosphere, the asthenosphere and the subasthenopsheric mantle. The obtained displacement, rotation, and strain fields are determined by the set of physical and mechanical properties and boundary conditions. The behavior of the parameters describes the state of stable equilibrium, which is characterized by the presence of curvature zones of the contour of the sphere, the resulting field of stress gradients, the predominance of horizontal displacements over vertical displacements, and the relationship between the strains at the surface and deep levels.