In materials science, a general rule of mixtures is a weighted mean used to predict various properties of a composite material .^[2][3][4] It provides a theoretical upper- and lower-bound on properties such as the elastic modulus, ultimate tensile strength, thermal conductivity, and electrical conductivity.^[4] In general there are two models, one for axial loading (Voigt model),^[3][5] and one for transverse loading (Reuss model).^[3][6]

In general, for some material property ${\displaystyle E}$ (often the elastic modulus^[2]), 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

It is a common mistake to believe that this is the upper-bound modulus for Young's modulus. The real upper-bound Young's modulus is larger than ${\displaystyle E_{c))$ given by this formula. Even if both constituents are isotropic, the real upper bound is ${\displaystyle E_{c))$ plus a term in the order of square of the difference of the Poisson's ratios of the two constituents.^[1]

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.^[3]

Derivation for elastic modulus

Voigt Modulus

Consider a composite material under uniaxial tension ${\displaystyle \sigma _{\infty ))$. If the material is to stay intact, the strain of the fibers, ${\displaystyle \epsilon _{f))$ must equal the strain of the matrix, ${\displaystyle \epsilon _{m))$. Hooke's law for uniaxial tension hence gives

${\displaystyle {\frac {\sigma _{f)){E_{f))}=\epsilon _{f}=\epsilon _{m}={\frac {\sigma _{m)){E_{m))))$

(1)

where ${\displaystyle \sigma _{f))$, ${\displaystyle E_{f))$, ${\displaystyle \sigma _{m))$, ${\displaystyle E_{m))$ are the stress and elastic modulus of the fibers and the matrix, respectively. Noting stress to be a force per unit area, a force balance gives that

${\displaystyle \sigma _{\infty }=f\sigma _{f}+\left(1-f\right)\sigma _{m))$

(2)

where ${\displaystyle f}$ is the volume fraction of the fibers in the composite (and ${\displaystyle 1-f}$ is the volume fraction of the matrix).

If it is assumed that the composite material behaves as a linear-elastic material, i.e., abiding Hooke's law ${\displaystyle \sigma _{\infty }=E_{c}\epsilon _{c))$ for some elastic modulus of the composite ${\displaystyle E_{c))$ and some strain of the composite ${\displaystyle \epsilon _{c))$, then equations 1 and 2 can be combined to give

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

Finally, since ${\displaystyle \epsilon _{c}=\epsilon _{f}=\epsilon _{m))$, the overall elastic modulus of the composite can be expressed as^[7]

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

Reuss modulus

Now let the composite material be loaded perpendicular to the fibers, assuming that ${\displaystyle \sigma _{\infty }=\sigma _{f}=\sigma _{m))$. The overall strain in the composite is distributed between the materials such that

${\displaystyle \epsilon _{c}=f\epsilon _{f}+\left(1-f\right)\epsilon _{m}.}$

The overall modulus in the material is then given by

${\displaystyle E_{c}={\frac {\sigma _{\infty )){\epsilon _{c))}={\frac {\sigma _{f)){f\epsilon _{f}+\left(1-f\right)\epsilon _{m))}=\left({\frac {f}{E_{f))}+{\frac {1-f}{E_{m))}\right)^{-1))$

since ${\displaystyle \sigma _{f}=E\epsilon _{f))$, ${\displaystyle \sigma _{m}=E\epsilon _{m))$.^[7]

Other properties

Similar derivations give the rules of mixtures for

• mass density:
  $${\displaystyle \rho _{c}=\rho _{f}\centerdot f+\rho _{M}\centerdot (1-f)}$$

  
  where f is the atomic percent of fiber in the mixture.
• ultimate tensile strength:
  $${\displaystyle \left({\frac {f}{\sigma _{UTS,f))}+{\frac {1-f}{\sigma _{UTS,m))}\right)^{-1}\leq \sigma _{UTS,c}\leq f\sigma _{UTS,f}+\left(1-f\right)\sigma _{UTS,m))$$

  
• thermal conductivity:
  $${\displaystyle \left({\frac {f}{k_{f))}+{\frac {1-f}{k_{m))}\right)^{-1}\leq k_{c}\leq fk_{f}+\left(1-f\right)k_{m))$$

  
• electrical conductivity:
  $${\displaystyle \left({\frac {f}{\sigma _{f))}+{\frac {1-f}{\sigma _{m))}\right)^{-1}\leq \sigma _{c}\leq f\sigma _{f}+\left(1-f\right)\sigma _{m))$$