On the Plane Strain and Plane Stress Solutions of Functionally Graded Rotating Solid Shaft and Solid Disk Problems

Abstract

Closed form solutions to functionally graded rotating solid shaft and rotating solid disk problems are obtained under generalized plane strain and plane stress assumptions, respectively. The nonhomogeneity in the material arises from the fact that the modulus of elasticity of the material varies radially according to two different continuously nonlinear forms: exponential and parabolic. Both forms contain two material parameters and lead to finite values of the modulus of elasticity at the center. Analytical expressions for the stresses at the center are determined. These limiting expressions indicate that at the center of shaft/disk: (i) the stresses are finite, (ii) the radial and the circumferential stress components are equal, and (iii) the values of the stresses are independent of the variation of the modulus of elasticity. It is also shown mathematically that the nonhomogeneous solutions presented here reduce to those of homogeneous ones by an appropriate choice of the material parameters describing the variation of the modulus of elasticity.