An effective general solution to the inhomogeneous spatial axisymmetric problem and its applications in functionally graded materials

The axisymmetric problem is a typical problem in the theory of elasticity, and the inhomogeneous spatial problem has an especially wider range of applications. In this paper, we first present an effective analytical elastic general solution to the inhomogeneous spatial axisymmetric problem. The specific descriptions of inhomogeneity are: Young's modulus is an arbitrary function of both radius and thickness coordinates, and Poisson's ratio is a constant. The elastic stress method is applied to obtain a general stress solution and a relatively concise analytical displacement solution, the degenerate forms of which are consistent with the existing results. Based on this general solution, we next study an axisymmetric bending problem of functionally graded circular plates subjected to a transverse loading q0r(n) (n is an even number) and give explicitly analytical elastic solutions to the case of uniform loadings under simply supported and two types of clamped supported conditions. The final analytic solutions correspond well to the existing numerical results. The distributions of explicit elastic fields related to the inhomogeneous parameter reveal the influence of inhomogeneity on the stress and displacement in FGM circular plates intuitively, which makes it possible to control the elastic performance of FGM plates more accurately.

Automated summary

This paper presents an effective analytical elastic general solution to the inhomogeneous spatial axisymmetric problem and studies the axisymmetric bending problem of functionally graded circular plates based on this general solution, obtaining analytical solutions consistent with existing numerical results. The explicit elastic field distributions related to the inhomogeneous parameter demonstrate the influence of inhomogeneity on stress and displacement in FGM circular plates.