Rule of mixtures explained

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. In general there are two models, one for axial loading (Voigt model),^[4] and one for transverse loading (Reuss model).^[5]

In general, for some material property

(often the elastic modulus), the rule of mixtures states that the overall property in the direction parallel to the fibers may be as high as where is the volume fraction of the fibers is the material property of the fibers is the material property of the matrixIn 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 If the property under study is the elastic modulus, this quantity is called the lower-bound modulus, and corresponds to a transverse loading.

Derivation for elastic modulus

Voigt Modulus

. If the material is to stay intact, the strain of the fibers, must equal the strain of the matrix, . Hooke's law for uniaxial tension hence giveswhere , , , 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 thatwhere is the volume fraction of the fibers in the composite (and 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

for some elastic modulus of the composite and some strain of the composite , then equations and can be combined to give Finally, since , the overall elastic modulus of the composite can be expressed as^[6]

Reuss modulus

Now let the composite material be loaded perpendicular to the fibers, assuming that

. The overall strain in the composite is distributed between the materials such that The overall modulus in the material is then given by since , .

Other properties

Similar derivations give the rules of mixtures for

• mass density:$$\backslash rho\_c=\backslash rho\_f\backslash centerdot\; f+\backslash rho\_M\backslash centerdot\; (1-f)$$ where f is the atomic percent of fiber in the mixture.
• ultimate tensile strength:$$\backslash left(\backslash frac\; +\; \backslash frac\backslash right)^\; \backslash leq\; \backslash sigma\_\; \backslash leq\; f\backslash sigma\_\; +\; \backslash left(1-f\backslash right)\backslash sigma\_$$
• thermal conductivity:$$\backslash left(\backslash frac\; +\; \backslash frac\backslash right)^\; \backslash leq\; k\_c\; \backslash leq\; fk\_f\; +\; \backslash left(1-f\backslash right)k\_m$$
• electrical conductivity:$$\backslash left(\backslash frac\; +\; \backslash frac\backslash right)^\; \backslash leq\; \backslash sigma\_c\; \backslash leq\; f\backslash sigma\_f\; +\; \backslash left(1-f\backslash right)\backslash sigma\_m$$

See also

When considering the empirical correlation of some physical properties and the chemical composition of compounds, other relationships, rules, or laws, also closely resembles the rule of mixtures:

• Amagat's law – Law of partial volumes of gases
• Gladstone–Dale equation – Optical analysis of liquids, glasses and crystals
• Kopp's law – Uses mass fraction
• Kopp–Neumann law – Specific heat for alloys
  • Richmann's law – Law for the mixing temperature Vegard's law – Crystal lattice parameters

External links

• Rule of mixtures calculator

Notes and References

1. Book: Alger, Mark. S. M.. Polymer Science Dictionary. 2nd. 1997. Springer Publishing. 0412608707.
2. Web site: Stiffness of long fibre composites. University of Cambridge. 1 January 2013.
3. Book: Askeland. Donald R.. Fulay. Pradeep P.. Wright. Wendelin J.. The Science and Engineering of Materials. 6th. 2010-06-21. Cengage Learning. 9780495296027.
4. Voigt. W.. Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper. Annalen der Physik. 1889. 274. 12 . 573–587. 10.1002/andp.18892741206. 1889AnP...274..573V .
5. Reuss. A.. Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Zeitschrift für Angewandte Mathematik und Mechanik. 1929. 9. 1 . 49–58. 10.1002/zamm.19290090104. 1929ZaMM....9...49R.
6. Web site: Derivation of the rule of mixtures and inverse rule of mixtures. University of Cambridge. 1 January 2013.