Tailoring of functionally graded spheres using a uniform stress condition

Abstract

Homogeneous and isotropic thick spheres loaded with constant internal and/or external pressures can not be economically designed because the maximum equivalent stress is a local value. It was analytically demonstrated that, by neglecting the body loads, a functionally graded material (FGM) may be characterized in linear static analyses by two material constants: Young's modulus E(r) and Poisson's ratio ν(r). If these two functions are both known, the solution of the problem, displacement and stress distributions may be relatively easy obtained. For the inverse problem, in which a desired stress combination distribution is imposed, finding of E(r) and ν(r) is more difficult, if such a solution exists. More than that, if the solution exists, it is not unique, because two unknown functions are involved. For ν(r) = const., analytical solutions are available for E(r), but only for two particular stress conditions. In this paper, the inverse problem is solved iteratively using a finite element model and an algorithm of stress uniformization developed by the authors of this paper is proposed. In this original approach, the existing solutions were reproduced as a verification and afterwards new solutions were obtained for the remaining classical theories of resistance. The new obtained solutions were also verified by using the analytical solutions of the direct problem.