Theoretical Models for Composite Material Response

Simple models

The Voigt model is as simple relationship between the moduli of the components and the effective modulus of the composite. (Briefly described in Reference 1)

E is the effective modulus. E1 and E2 are moduli of the individual components and v1 and v2 are the volume fractions.

This is a simple mixing rule for an iso-strain state. This model assumes that the reinforcement is aligned in a single direction and that each component experiences the same strain.

Ruess later modified this formulations to create the following (Reference 2)

These two simple models form upper and lower bounds on the elastic modulus as determined by Hill (Reference 4).

More restrictive bounds were calulated by Hashin and Shtrikman (Reference 5) as the following:

Lower Bound: $K^{*} \geq K_{1} + \frac{v_{2}}{\frac{1}{K_{2}-K_{1}}+\frac{3*v_{1}}{3K_{1}+4*G_{1}}}$

Upper Bound: $K^{*} \leq K_{2} + \frac{v_{1}}{\frac{1}{K_{1}-K_{2}}+\frac{3*v_{2}}{3K_{2}+4*G_{2}}}$

Here K is the bulk modulus which is related to the Lame constants by the following equation

These bounds do not take into account actual microstructures of the samples simply their volume fractions.

http://onlinelibrary.wiley.com/doi/10.1002/pc.20002/pdf