Elastic Moduli of Thickly Coated Particle and Fiber-Reinforced Composites

Based on the models of Hashin (1962) and Hashin and Rosen (1964), the effective elastic moduli of thickly coated particle and fiber-reinforced composites are derived. The microgeometry of the composite is that of a progressively filled composite sphere or cylinder element model. The exact solutions of the effective bulk modulus κ of the particle-reinforced composite and those of the plain-strain bulk modulus κ₂₃, axial shear modulus μ₁₂, longitudinal Young’s modulus E₁₁, major Poisson ratio ν₁₂, of the fiber-reinforced one are derived by the replacement method. The bounds for the effective shear modulus μ and the effective transverse shear modulus μ₂₃ of these two kinds of composite, respectively, are solved with the aid of Christensen and Lo’s (1979) formulations. By considering the six possible geometrical arrangements of the three constituent phases, the values of κ, and of κ₂₃, μ₁₂, E₁₁, and ν₁₂ are found to always lie within the Hashin-Shtrikman (1963) bounds, and the Hashin (1965), Hill (1964), and Walpole (1969) bounds, respectively, but unlike the two-phase composites, none coincides with their bounds. The bounds of μ and μ₂₃ derived here are consistently tighter than their bounds but, as for the two-phase composites, one of the bounds sometimes may fall slightly below or above theirs and therefore it is suggested that these two sets of bounds be used in combination, always choosing the higher for the lower bound and the lower for the upper one.