Effective bulk moduli of two functionally graded composites

This paper considers the effective bulk moduli and the underlying elastic fields of a particle- and a fiber-reinforced composite whose matrix properties are graded linearly along the radial distance. The governing Fuchsian equations are first transformed into Riemann equations which, by means of change of the dependent variable, are further transformed into hypergeometric equations which are solved in terms of the hypergeometric function. This set of solutions provides an explicit dependence of the effective bulk behavior of the composites with arbitrary variations of the slope of the matrix moduli.