On eshelby's s-tensor in a three-phase cylindrically concentric solid, and the elastic moduli of fiber-reinforced composites

The elastic field in a three-phase cylindrically concentric, transversely isotropic solid due to a uniform stress-free transformation strain in the central fiber is derived. The transformation strains considered are selected to include the uniaxial tension, plane-strain dilatation, transverse shear, and axial shear, and it is found that, with the exception of the transverse-shear condition, the strain fields in the fiber are also uniform. These solutions enable one to establish the five non-vanishing components of the average S-tensor in such a three-phase solid. With the help of these components, the modified Mori-Tanaka method recently suggested by Luo and Weng (1987) is applied to calculate the five elastic moduli of a fiber-reinforced composite. Surprisingly enough four of the five moduli are found to remain unchanged under this modification; they all coincide with the Hill-Hashin bounds. The fifth one-the transverse shear modulus-becomes stiffer than the original prediction (or Hashin's lower bound), but still lies below Christensen-Lo's result and Hashin's upper bound.