The influence of moduli slope of a linearly graded matrix on the bulk moduli of some particle- and fiber-reinforced composites

In the context of a functionally-graded composite whose matrix property varies linearly from the interface, exact analytical solutions are derived for the dependence of the effective bulk moduli of some particle- and fiber-reinforced composites on the moduli slope of the matrix phase. Specifically four types of property variation in the matrix are considered. In the particle case: (i) the slopes of the matrix properties emanate from the origin, (ii) the bulk modulus of the matrix varies linearly while its shear modulus remains constant, (iii) the shear modulus varies linearly while its bulk modulus remains constant, and (iv) incompressible matrix. In the fiber case it is the plane-strain bulk modulus of the matrix instead of the ordinary 3-dimensional bulk modulus that varies. Unlike the series representations or numerical solutions, a novel feature of the results here is that they are given in simple analytical forms, rendering them useful for future applications. Some calculations are presented to illustrate the influence of the slope of the modulus variation on the overall properties of the composites, and a step is also offered to construct such linearly graded matrix properties.