Rule of mixtures

In materials science, a general rule of mixtures is a weighted mean used to predict various properties of a composite material . It provides a theoretical upper- and lower-bound on properties such as the elastic modulus, ultimate tensile strength, thermal conductivity, and electrical conductivity. In general there are two models, the rule of mixtures for axial loading (Voigt model), and the inverse rule of mixtures for transverse loading (Reuss model).

For some material property ${\displaystyle E}$, the rule of mixtures states that the overall property in the direction parallel to the fibers could be as high as

${\displaystyle E_{\parallel }=fE_{f}+\left(1-f\right)E_{m}}$

The inverse rule of mixtures states that in the direction perpendicular to the fibers, the elastic modulus of a composite could be as low as

${\displaystyle E_{\perp }=\left({\frac {f}{E_{f}}}+{\frac {1-f}{E_{m}}}\right)^{-1}.}$

where

• ${\displaystyle f={\frac {V_{f}}{V_{f}+V_{m}}}}$ is the volume fraction of the fibers
• ${\displaystyle E_{\parallel }}$ is the material property of the composite parallel to the fibers
• ${\displaystyle E_{\perp }}$ is the material property of the composite perpendicular to the fibers
• ${\displaystyle E_{f}}$ is the material property of the fibers
• ${\displaystyle E_{m}}$ is the material property of the matrix

If the property under study is the elastic modulus, these properties are known as the upper-bound modulus, corresponding to loading parallel to the fibers; and the lower-bound modulus, corresponding to transverse loading.