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, [1] ultimate tensile strength, thermal conductivity, and electrical conductivity. [3] In general there are two models, the rule of mixtures for axial loading (Voigt model), [2] [4] and the inverse rule of mixtures for transverse loading (Reuss model). [2] [5]
For some material property , the rule of mixtures states that the overall property in the direction parallel to the fibers could be as high as
The inverse rule of mixtures states that in the direction perpendicular to the fibers, the property could be as low as
where
If the property under study is the elastic modulus, these properties are known as the upper-bound modulus, corresponding to loading parallel to the fibers; and the lower-bound modulus, corresponding to transverse loading. [2]
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
1 |
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
2 |
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 parallel to the fibres 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 [6]
Assuming the Poisson's ratio of the two materials is the same, this represents the upper bound of the composite's elastic modulus. [7]
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 , . [6]
Similar derivations give the rules of mixtures for
A generalized equation for any loading condition between isostrain and isostress can be written as: [8]
where k is a value between 1 and −1.
For a composite containing a mixture of n different materials, each with a material property and volume fraction , where
then the rule of mixtures can be shown to give:
and the inverse rule of mixtures can be shown to give:
Finally, generalizing to some combination of the rule of mixtures and inverse rule of mixtures for an n-component system gives:
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: