Rule of mixtures
In materials science, a general rule of mixtures is a weighted mean used to predict various properties of a composite material . 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 (Voigt model), and one for transverse loading (Reuss model). In general, for some material property formula_1 (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 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 formula_6 given by this formula. Even if both constituents are isotropic, the real upper bound is formula_6 plus a term in the order of square of the difference of the Poisson's ratios of the two constituents. 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 formula_9. If the material is to stay intact, the strain of the fibers, formula_10 must equal the strain of the matrix, formula_11. Hooke's law for uniaxial tension hence gives where formula_12, formula_4, formula_14, formula_5 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 formula_16 is the volume fraction of the fibers in  the composite (and formula_17 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 formula_18 for some elastic modulus of the composite formula_6 and some strain of the composite formula_20, then equations and can be combined to give Finally, since formula_22, 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 formula_24. The overall strain in the composite is distributed between the materials such that The overall  modulus in the material is then given by since formula_27, formula_28. Other properties. Similar derivations give the rules of mixtures for 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: