Rule of mixtures

In materials science, a general rule of mixtures is a weighted mean used to predict various properties of a composite material .^[1][2][3] It provides a theoretical upper- and lower-bound on properties such as the elastic modulus, ultimate tensile strength, thermal conductivity, and electrical conductivity.^[3] In general there are two models, one for axial loading (Voigt model),^[2][4] and one for transverse loading (Reuss model).^[2][5]

In general, for some material property ${\displaystyle E}$ (often the elastic modulus^[1]), the rule of mixtures states that the overall property in the direction parallel to the fibers may be as high as

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

where

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

In the case of the elastic modulus, this is known as the upper-bound modulus, and corresponds to loading parallel to the fibers. The inverse rule of mixtures states that in the direction perpendicular to the fibers, the elastic modulus of a composite can be as low as

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

If the property under study is the elastic modulus, this quantity is called the lower-bound modulus, and corresponds to a transverse loading.^[2]