ANALYTICAL SOLUTIONS FOR DISPLACEMENTS AND STRESSES IN FUNCTIONALLY GRADED THICK-WALLED SPHERES SUBJECTED TO A UNIDIRECTIONAL OUTER TENSION

摘要

In the context of infinitesimal theory of elasticity, we derived analytical solutions for displacements and stresses in functionally graded thick-walled spheres under the application of a uniaxial outer tension. While the shear modulus in the graded sphere is allowed to vary as a power-law function of radial coordinate, the Poisson's ratio is treated as a constant. The semiinverse method of elasticity is first employed for proposing correct function forms of the radial and longitudinal displacements. The elastostatic Navier's equations of the power-law graded sphere lead to a system of second-order differential equations of the Euler type. The order is then reduced and the system is recast into a first-order differential matrix equation. Analytical solutions are subsequently developed by the coupling of differential equation and eigenvalue theories. Successfully solving this particular problem provides a valid analytical solution scheme for exploring elastic fields in graded hollow spheres subjected to nonhydrostatic boundary loads. In order to examine the effects of the power-law gradation and the radii ratio of the thick-walled sphere on stress distributions and stress concentration factors, extensive parametric studies are conducted. Analytical solutions of the graded thick-walled sphere are further compared with those of the homogeneous case as well as with the numerical results due to finite element modelings. The obtained results show that the property gradation significantly affects stress distributions through the thickness direction of the graded thick-walled sphere. When the shear modulus is designed as an increasing function of the radial coordinate, the high stress zone conventionally occurring near the inner boundary of homogeneous thick-walled spheres tends to shift toward to the outer surface vicinity. For a given radii ratio, an optimal power-law gradation leading to the lowest stress concentration factor can always be identified. The proposed method of solution and the obtained results are useful for the design and manufacturing of better performing spherical vessels.