Construction of homogeneous solutions in the torsion problem for a transversally isotropic sphere with variable elastic moduli

The object of research is the problem of torsion for a radially inhomogeneous transversally isotropic sphere and the study based on this three-dimensional stress-strain state. To establish the scope of applicability of existing applied theories and to create more refined applied theories of inhomogeneous shells, it is important to study the stress-strain state of inhomogeneous bodies based on three-dimensional equations of elasticity theory. The problem of torsion of a radially inhomogeneous transversally isotropic non-closed sphere containing none of the poles 0 and π is considered. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the lateral surface of the sphere is free from stresses, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The formulated boundary value problem is reduced to a spectral problem. After fulfilling the homogeneous boundary conditions specified on the side surfaces of the sphere, a characteristic equation is obtained with respect to the spectral parameter. The corresponding solutions are constructed depending on the roots of the characteristic equation. It is shown that the solution corresponding to the first group of roots is penetrating, and the stress state determined by this solution is equivalent to the torques of the stresses acting in an arbitrary section θ=const. The solutions corresponding to the countable set of the second group of roots have the character of a boundary layer localized in conic slices. In the case of significant anisotropy, some boundary layer solutions decay weakly and can cover the entire region occupied by the sphere. On the basis of the performed three-dimensional analysis, new classes of solutions (solutions having the character of a boundary layer) are obtained, which are absent in applied theories. In contrast to an isotropic radially inhomogeneous sphere, for a transversely isotropic radially inhomogeneous sphere, a weakly damped boundary layer solution appears, which can penetrate deep far from the conical sections and change the picture of the stress-strain state.