The axisymmetric problem of the theory of elasticity for the radially heterogeneous transverse‐isotopic nonclosed spheres is studied, which does not contain any of the poles 0 and π. The elasticity modules are taken as the linear functions of the radius of the sphere. It is assumed that the lateral surface of the sphere is free from stresses, and in the conical sections, the arbitrary stresses are set that provide equilibrium for the sphere. After consideration of the homogeneous boundary conditions, set on the lateral surfaces of the sphere, the characteristic equation for the spectral parameter is obtained. On the basis of the asymptotic analysis, a classification of the roots of the characteristic equation is made relatively small parameter that characterizes the thickness of the sphere. Corresponding asymptotic solutions are constructed depending on the roots of the characteristic equation. The behavior of the constructed solutions is studied in the internal parts of the sphere, as well as in the neighborhood of the conical sections.

Introduction. The paper considers an axisymmetric problem of elasticity theory for a radially inhomogeneous transversally isotopic nonclosed sphere containing none of the 0 and 𝜋 poles. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the side surface of the sphere is fixed, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The work objective is an asymptotic analysis of the problem of elasticity theory for a radially inhomogeneous transversally isotropic sphere of small thickness, and a study of a three-dimensional stress-strain state based on this analysis.Materials and Methods. The three-dimensional stress-strain state is investigated on the basis of the equations of elasticity theory by the method of homogeneous solutions and asymptotic analysis.Research Results. After the homogeneous boundary conditions set on the side surfaces of the sphere are met, a characteristic equation is obtained, and its roots are classified with respect to a small parameter characterizing the thickness of the sphere. The corresponding asymptotic solutions depending on the roots of the characteristic equation are constructed. It is shown that the solutions corresponding to a countable set of roots have the character of a boundary layer localized in conic slices. The branching of the roots generates new solutions that are characteristic only for a transversally isotropic radially inhomogeneous sphere. A weakly damping boundary layer solution appears, which can penetrate deep away from the conical sections and change the picture of the stress-strain state.Discussion and Conclusions. Based on the solutions constructed, it is possible to determine the applicability areas of existing applied theories and propose a new more refined applied theory for a radially inhomogeneous transversally isotropic spherical shell.