A doubly embedded self-consistent field micromechanics model is used to determine the effective mechanical properties of the composite tow as a function of the properties of the resin and fiber and composite volume fraction. This model, [Hill 1965] considers a single fiber embedded in a micro region of continuous matrix phase as shown below. This composite element is then embedded in a transversely isotropic homogeneous medium with properties matching the effective properties of the composite material. This model thus gives a closed from solution for the effective composite properties and expansion strains. The cross section of the fiber is assumed constant along its length.

Principle material directions in a unidirectional composite (left) and the fiber and matrix embedded in a homogeneous medium (right).

The simplifications introduced by the embedded cylinder model tend to mask the extent to which the neighboring fibers mutually influence their response characteristics. Therefore the relationships developed in this model tend to underestimate properties for high volume fraction fiber composites. The model also does not take the packing geometry into account.

The development of these equations are omitted for the sake of brevity but can be found in the literature. In the following equations, the 1, 2 and 3 directions are referred to as the L, T, and T, as shown in the figure above. The subscripts m and f correspond to the matrix and fiber properties, respectfully. Vf is the fiber volume fraction of the composite and k is the isotropic plane strain bulk modulus defined by: 

               

From his work the longitudinal Young’s modulus is given by:

               

and the transverse Young’s modulus is given by

               

kT is the effective plane strain bulk modulus of the composite and is given by:

               

The major Poisson’s Ration is given by:

               

The in-plane shear modulus is given by:

               

The transverse shear modulus is found from the following:

               

The transverse Poisson’s ratio is given by: