Rule of mixtures

The upper and lower bounds on the elastic modulus of a composite material, as predicted by the rule of mixtures. The actual elastic modulus lies between the curves.

In materials science, a general rule of mixtures is a weighted mean used to predict various properties of a composite material made up of continuous and unidirectional fibers.^[1][2][3] It provides a theoretical upper- and lower-bound on properties such as the elastic modulus, mass density, 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 $E$ (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

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

where

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

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

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

Derivation for elastic modulus

Upper-bound modulus

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

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

(1)

where $\sigma _{f}$, $E_{f}$, $\sigma _{m}$, $E_{m}$ 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

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

(2)

where $f$ is the volume fraction of the fibers in the composite (and $1-f$ 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 $\sigma _{\infty }=E_{c}\epsilon _{c}$ for some elastic modulus of the composite $E_{c}$ and some strain of the composite $\epsilon _{c}$, then equations 1 and 2 can be combined to give

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

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

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

Lower-bound modulus

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

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

The overall modulus in the material is then given by

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

since $\sigma _{f}=E\epsilon _{f}$, $\sigma _{m}=E\epsilon _{m}$.^[6]

Other properties

Similar derivations give the rules of mixtures for

• mass density

$\left({\frac {f}{\rho _{f}}}+{\frac {1-f}{\rho _{m}}}\right)^{{-1}}\leq \rho _{c}\leq f\rho _{f}+\left(1-f\right)\rho _{m}$

• ultimate tensile strength

$\left({\frac {f}{\sigma _{{UTS,f}}}}+{\frac {1-f}{\sigma _{{UTS,m}}}}\right)^{{-1}}\leq \sigma _{{UTS,c}}\leq f\sigma _{{UTS,f}}+\left(1-f\right)\sigma _{{UTS,m}}$

• thermal conductivity

$\left({\frac {f}{k_{f}}}+{\frac {1-f}{k_{m}}}\right)^{{-1}}\leq k_{c}\leq fk_{f}+\left(1-f\right)k_{m}$

• electrical conductivity

$\left({\frac {f}{\sigma _{f}}}+{\frac {1-f}{\sigma _{m}}}\right)^{{-1}}\leq \sigma _{c}\leq f\sigma _{f}+\left(1-f\right)\sigma _{m}$

References

1. Alger, Mark. S. M. (1997). Polymer Science Dictionary (2nd ed.). Springer Publishing. ISBN 0412608707.
2. "Stiffness of long fibre composites". University of Cambridge. Retrieved 1 January 2013.
3. Askeland, Donald R.; Fulay, Pradeep P.; Wright, Wendelin J. (2010-06-21). The Science and Engineering of Materials (6th ed.). Cengage Learning. ISBN 9780495296027.
4. Voigt, W. (1889). "Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper". Annalen der Physik. 274: 573–587. Bibcode:1889AnP...274..573V. doi:10.1002/andp.18892741206.
5. Reuss, A. (1929). "Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle". Zeitschrift für Angewandte Mathematik und Mechanik. 9: 49–58. doi:10.1002/zamm.19290090104.
6. "Derivation of the rule of mixtures and inverse rule of mixtures". University of Cambridge. Retrieved 1 January 2013.

External links

• Rule of mixtures calculator