Physics:Rule of mixtures

Short description: Relation between properties and composition of a compound

The upper and lower bounds on the elastic modulus of a composite material assuming equal Poisson's coefficients.

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, ultimate tensile strength, thermal conductivity, and electrical conductivity. In general there are two models, the rule of mixtures for axial loading (Voigt model), and the inverse rule of mixtures for transverse loading (Reuss model).

For some material property $E$, the rule of mixtures states that the overall property in the direction parallel to the fibers could be as high as

$E_{\parallel }=f{E}_f+(1-f)E_m$

The inverse rule of mixtures states that in the direction perpendicular to the fibers, the property could be as low as

$E_{\perp }={(\frac{f}{E_f}+\frac{1-f}{E_m})}^{-1}.$

where

• $f=\frac{V_f}{V_f+V_m}$ is the volume fraction of the fibers
• $E_{\parallel }$ is the material property of the composite parallel to the fibers
• $E_{\perp }$ is the material property of the composite perpendicular to the fibers
• $E_f$ is the material property of the fibers
• $E_m$ is the material property of the matrix

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.

Derivation for elastic modulus

Rule of mixtures / Voigt model / equal strain

Consider a composite material under uniaxial tension ${\sigma }_{\mathrm{\infty }}$. If the material is to stay intact, the strain of the fibers, ${ϵ}_f$ must equal the strain of the matrix, ${ϵ}_m$. Hooke's law for uniaxial tension hence gives

$\frac{{\sigma }_f}{E_f}={ϵ}_f={ϵ}_m=\frac{{\sigma }_m}{E_m}$

(1)

where ${\sigma }_f$, $E_f$, ${\sigma }_m$, $E_m$ 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

${\sigma }_{\mathrm{\infty }}=f{\sigma }_f+(1-f){\sigma }_m$

(2)

where $f$ is the volume fraction of the fibers in the composite (and $1-f$ 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 ${\sigma }_{\mathrm{\infty }}=E_{\parallel }{ϵ}_c$ for some elastic modulus of the composite parallel to the fibres $E_{\parallel }$ and some strain of the composite ${ϵ}_c$, then equations 1 and 2 can be combined to give

$E_{\parallel }{ϵ}_c=f{E}_f{ϵ}_f+(1-f)E_m{ϵ}_m.$

Finally, since ${ϵ}_c={ϵ}_f={ϵ}_m$, the overall elastic modulus of the composite can be expressed as

$E_{\parallel }=f{E}_f+(1-f)E_m.$

Assuming the Poisson's ratio of the two materials is the same, this represents the upper bound of the composite's elastic modulus.

Inverse rule of mixtures / Reuss model / equal stress

Now let the composite material be loaded perpendicular to the fibers, assuming that ${\sigma }_{\mathrm{\infty }}={\sigma }_f={\sigma }_m$. The overall strain in the composite is distributed between the materials such that

${ϵ}_c=f{ϵ}_f+(1-f){ϵ}_m.$

The overall modulus in the material is then given by

$E_{\perp }=\frac{{\sigma }_{\mathrm{\infty }}}{{ϵ}_c}=\frac{{\sigma }_f}{f{ϵ}_f+(1-f){ϵ}_m}={(\frac{f}{E_f}+\frac{1-f}{E_m})}^{-1}$

since ${\sigma }_f=E_f{ϵ}_f$, ${\sigma }_m=E_m{ϵ}_m$.

Other properties

Similar derivations give the rules of mixtures for

• mass density:$${(\frac{f}{{\rho }_f}+\frac{1-f}{{\rho }_c})}^{-1}\le {\rho }_f\cdot f+{\rho }_M\cdot (1-f)$$ where f is the atomic percent of fiber in the mixture.
• ultimate tensile strength:$${(\frac{f}{{\sigma }_{UTS,f}}+\frac{1-f}{{\sigma }_{UTS,m}})}^{-1}\le {\sigma }_{UTS,c}\le f{\sigma }_{UTS,f}+(1-f){\sigma }_{UTS,m}$$
• thermal conductivity:$${(\frac{f}{k_f}+\frac{1-f}{k_m})}^{-1}\le k_c\le f{k}_f+(1-f)k_m$$
• electrical conductivity:$${(\frac{f}{{\sigma }_f}+\frac{1-f}{{\sigma }_m})}^{-1}\le {\sigma }_c\le f{\sigma }_f+(1-f){\sigma }_m$$

Generalizations

Some proportion of rule of mixtures and inverse rule of mixtures

A generalized equation for any loading condition between isostrain and isostress can be written as:

$(E_c{)}^k=f(E_f{)}^k+(1-f)(E_m{)}^k$

where k is a value between 1 and −1.

More than 2 materials

For a composite containing a mixture of n different materials, each with a material property $E_i$ and volume fraction $V_i$, where

$\sum _{i=1}^n{V}_i=1,$

then the rule of mixtures can be shown to give:

$E_c=\sum _{i=1}^n{V}_i{E}_i$

and the inverse rule of mixtures can be shown to give:

$\frac{1}{E_c}=\sum _{i=1}^n\frac{V_i}{E_i}.$

Finally, generalizing to some combination of the rule of mixtures and inverse rule of mixtures for an n-component system gives:

$(E_c{)}^k=\sum _{i=1}^n{V}_i(E_i{)}^k$

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 – Heat capacity, with f as the mass fraction
• Richmann's law – Law for the mixing temperature
• Vegard's law – Crystal lattice parameters

External links

• Rule of mixtures calculator