Rule of mixtures

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, mass density, 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]

The upper and lower bounds on the elastic modulus of a composite material with the lower bound predicted by the rule of mixtures given here but the upper bound larger than the rule of mixtures given here.[1] The actual elastic modulus lies between the curves.

In general, for some material property (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


  • is the volume fraction of the fibers
  • is the material property of the fibers
  • 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 given by this formula. Even if both constituents are isotropic, the real upper bound is 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

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 modulusEdit

Upper-bound modulusEdit

Consider a composite material under uniaxial tension  . 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 gives







where  ,  ,  ,   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







where   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 1 and 2 can be combined to give


Finally, since  , the overall elastic modulus of the composite can be expressed as[7]


Lower-bound modulusEdit

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  ,  .[7]

Other propertiesEdit

Similar derivations give the rules of mixtures for

  • mass density:
  • ultimate tensile strength:
  • thermal conductivity:
  • electrical conductivity:

See alsoEdit

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:


  1. ^ a b Yu, Wenbin (2016). An Introduction to Micromechanics. Switzerland: Trans Tech Publications. pp. 3–24. ISBN 9783038357469.
  2. ^ a b Alger, Mark. S. M. (1997). Polymer Science Dictionary (2nd ed.). Springer Publishing. ISBN 0412608707.
  3. ^ a b c d "Stiffness of long fibre composites". University of Cambridge. Retrieved 1 January 2013.
  4. ^ a b Askeland, Donald R.; Fulay, Pradeep P.; Wright, Wendelin J. (2010-06-21). The Science and Engineering of Materials (6th ed.). Cengage Learning. ISBN 9780495296027.
  5. ^ Voigt, W. (1889). "Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper". Annalen der Physik. 274 (12): 573–587. Bibcode:1889AnP...274..573V. doi:10.1002/andp.18892741206.
  6. ^ Reuss, A. (1929). "Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle". Zeitschrift für Angewandte Mathematik und Mechanik. 9 (1): 49–58. Bibcode:1929ZaMM....9...49R. doi:10.1002/zamm.19290090104.
  7. ^ a b "Derivation of the rule of mixtures and inverse rule of mixtures". University of Cambridge. Retrieved 1 January 2013.

External linksEdit