Rule of mixtures


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. 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. In general there are two models, one for axial loading, and one for transverse loading.
In general, for some material property , the rule of mixtures states that the overall property in the direction parallel to the fibers may be as high as
where
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
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

Upper-bound modulus

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.
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

Lower-bound 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