# Tailoring of functionally graded spheres using a uniform stress condition

## Keywords:

thick spheres, elastic analysis, functional graded material (FGM), finite element analysis (FEA), full stressed design (FSD), iterative technique## 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 *v*(*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 *v*(*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 *v*(*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.

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