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