Thermal stresses and effective thermal expansion coefficient of a functionally gradient sphere

We present the exact solution to the problem of uniform heating of a spherical body whose elastic moduli and thermal expansion coefficient vary linearly with the radius. The Frobenius series method is used to find exact expressions for the displacements and stresses. Both the radial and hoop stresses are largest in magnitude at the center of the sphere. The radial stress decays to zero at the outer edge, whereas the hoop stresses always change sign at some intermediate value of the radius. We also find an exact expression for the effective thermal expansion coefficient. For the special case where the thermal expansion coefficient varies with the radius but the elastic moduli are uniform, the effective thermal expansion coefficient of the sphere is equal to the volumetric average of the local thermal expansion coefficient. In the more general case, where the moduli also vary the moduli variations have very little influence on the effective thermal expansion coefficient.