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,^[1] thermal conductivity, and electrical conductivity.^[3]

In general there are two models. The rule of mixtures (the Voigt model) is derived under the assumption that the strain in both constituents is equal. ^[2]^[4] The inverse rule of mixtures (the Reuss model) is found if the stress in both constituents is assumed equal. ^[2]^[5] Respectively, these could model axial- and transverse loading in a fiber-reinforced composite material.

For the Young's modulus $E$ , the rule of mixtures states that the overall modulus equals

{\overline {E}}_{V}=fE_{1}+\left(1-f\right)E_{2}

.

The inverse rule of mixtures states that Young's modulus equals

{\overline {E}}_{R}=\left({\frac {f}{E_{1}}}+{\frac {1-f}{E_{2}}}\right)^{-1}.

where

$f={\frac {V_{1}}{V_{1}+V_{2}}}$ is the volume fraction of the first constituent
$E_{1},E_{2}$ are the stiffness of constituents 1 and 2 respectively
${\overline {E}}_{V},{\overline {E}}_{R}$ are the stiffnesses homogenized according to the Voigt and Reuss assumptions respectively

These two moduli are often treated as an upper- and lower bound, with the actual modulus lying somewhere inbetween.

Derivation for elastic modulus

Rule of mixtures / Voigt model / equal strain

Consider a composite material under uniaxial tension $\sigma _{\infty }$ . Under the Voigt assumption, we model the strain in the two constituents as equal. In the context of fiber-reinforced composites, one might interpret this as an applied strain along the fiber direction. Hooke's law for uniaxial tension gives

{\frac {\sigma _{1}}{E_{1}}}=\epsilon _{1}={\overline {\epsilon }}=\epsilon _{2}={\frac {\sigma _{2}}{E_{2}}}

1

where $\sigma _{1}$ and $\sigma _{2}$ are the stresses of constituents 1 and 2 respectively, and we define the homogenized strain as ${\overline {\epsilon }}$ . Noting stress to be a force per unit area, a force balance gives that

\sigma _{\infty }=f\sigma _{1}+\left(1-f\right)\sigma _{2}

2

Equations 1 and 2 can be combined to give

{\overline {E}}_{V}{\overline {\epsilon }}=fE_{1}\epsilon _{1}+\left(1-f\right)E_{2}\epsilon _{2}.

Finally, since ${\overline {\epsilon }}=\epsilon _{1}=\epsilon _{2}$ , the overall elastic modulus of the composite can be expressed as^[6]

{\overline {E}}_{V}=fE_{1}+\left(1-f\right)E_{2}.

Assuming the Poisson's ratio of the two materials is the same, this represents the upper bound of the composite's elastic modulus.^[7]

Inverse rule of mixtures / Reuss model / equal stress

Alternatively, we can assume that the stress in the two constituents is equal, i.e. $\sigma _{\infty }=\sigma _{1}=\sigma _{2}$ . This corresponds to the two constituents being loaded in series, which in the context of fiber-reinforced composites roughly corresponds to transverse loading. In this case, the overall strain is distributed according to

{\overline {\epsilon }}=f\epsilon _{1}+\left(1-f\right)\epsilon _{2}.

The overall modulus in the material is then given by

{\overline {E}}_{R}={\frac {\sigma _{\infty }}{\overline {\epsilon }}}={\frac {\sigma _{1}}{f\epsilon _{1}+\left(1-f\right)\epsilon _{2}}}=\left({\frac {f}{E_{1}}}+{\frac {1-f}{E_{2}}}\right)^{-1}

since $\sigma _{1}=E_{1}\epsilon _{1}$ , $\sigma _{2}=E_{2}\epsilon _{2}$ .^[6]

Other properties

Similar derivations give the rules of mixtures for

mass density: $\left({\frac {f}{\rho _{1}}}+{\frac {1-f}{\rho _{2}}}\right)^{-1}\leq \ {\overline {\rho }}\leq f\rho _{1}+(1-f)\rho _{2}$

thermal conductivity: $\left({\frac {1}{k_{1}}}+{\frac {1-f}{k_{2}}}\right)^{-1}\leq {\overline {k}}\leq fk_{1}+\left(1-f\right)k_{2}$

electrical conductivity: $\left({\frac {f}{\sigma _{1}}}+{\frac {1-f}{\sigma _{2}}}\right)^{-1}\leq {\overline {\sigma }}\leq f\sigma _{1}+\left(1-f\right)\sigma _{2}$

Notably, these forms are typically not applicable to strength-related properties. Consider the case where the strain in both phases is equal. Then, the rule of mixtures would only be applicable when both materials fail at the same strain. If this is not the case, the least ductile material will fail first. Thus, the strength of the composite would be determined not as a mix of constituent strengths, but rather as a function of the strength of a single constituent. In turn, this supposes that another failure mode, such as debonding, does not occur before either phase fails.