Inverse problems of the inhomogeneous theory of elasticity for thick-walled shells

Abstract

The inhomogeneous theory of elasticity considers bodies, the mechanical characteristics of which (the modulus of elasticity and Poisson’s ratio) are functions of the coordinates. If indirect problems of the inhomogeneous theory of elasticity are identifi ed, and the stress-strain state of the body has well-known functions of mechanical characteristics, the essence of inverse problems is to determine the functions of the inhomogeneity for a given stress state of the body. One of the fi rst solutions to such an inverse problem was published in the work of Lekhnitskii (“Radial distribution of stresses in the wedge and half-plane with variable modulus of elasticity”. PMM, XXVI(1), pp. 146–151, 1962). In this article, we consider one-dimensional inverse problems for thick-walled cylindrical and spherical shells that are subjected to internal and external pressures in a non-varying temperature fi eld. The aim of this work is to identify the dependence of the elastic modulus on the radial co-ordinate for which the equivalent stress according to a particular theory of strength will be constant at all points of the body (such structures are called equal stress), or the equivalent stress in all points will be equal to the strength of the material (such structures are called equal strength). For example, the author has proven that the limit