Compression of a sphere with radially varying elastic moduli

We present the exact solution to the problem of hydrostatic compression of a spherical body whose elastic moduli vary linearly with radius. The solution is developed as a power series in the radius r, and in ε{lunate}, a parameter that quantifies the degree of elastic inhomogeneity of the sphere. In contrast to a homogeneous sphere, the stresses and strains are found to vary with the radius. We also find the effective bulk modulus of the inhomogeneous sphere, and compare it to the predictions of the Voigt, Reuss and Hashin-Shtrikman bounds. This solution should be of use in the analysis of functionally gradient materials.