Analysis of the stress-strain state of a radially inhomogeneous transversely isotropic sphere with a fixed side surface

Abstract

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.