Rule of mixtures

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 . ^{[1] [2] [3]} It provides a theoretical upper- and lower-bound on properties such as the elastic modulus, ^[1] thermal conductivity, and electrical conductivity. ^[3]

In general there are two models. The rule of mixtures (the Voigt model) is derived under the assumption that the strain in both constituents is equal. ^{[2] [4]} The inverse rule of mixtures (the Reuss model) is found if the stress in both constituents is assumed equal. ^{[2] [5]} Respectively, these could model axial- and transverse loading in a fiber-reinforced composite material.

For the Young's modulus ${\displaystyle E}$, the rule of mixtures states that the overall modulus equals

${\displaystyle {\overline {E}}_{V}=fE_{1}+\left(1-f\right)E_{2}}$.

The inverse rule of mixtures states that Young's modulus equals

${\displaystyle {\overline {E}}_{R}=\left({\frac {f}{E_{1}}}+{\frac {1-f}{E_{2}}}\right)^{-1}.}$

where

• ${\displaystyle f={\frac {V_{1}}{V_{1}+V_{2}}}}$ is the volume fraction of the first constituent
• ${\displaystyle E_{1},E_{2}}$ are the stiffness of constituents 1 and 2 respectively
• ${\displaystyle {\overline {E}}_{V},{\overline {E}}_{R}}$ are the stiffnesses homogenized according to the Voigt and Reuss assumptions respectively

These two moduli are often treated as an upper- and lower bound, with the actual modulus lying somewhere inbetween.

Derivation for elastic modulus

Rule of mixtures / Voigt model / equal strain

Consider a composite material under uniaxial tension ${\displaystyle \sigma _{\infty }}$. Under the Voigt assumption, we model the strain in the two constituents as equal. In the context of fiber-reinforced composites, one might interpret this as an applied strain along the fiber direction. Hooke's law for uniaxial tension gives

${\displaystyle {\frac {\sigma _{1}}{E_{1}}}=\epsilon _{1}={\overline {\epsilon }}=\epsilon _{2}={\frac {\sigma _{2}}{E_{2}}}}$

1

where ${\displaystyle \sigma _{1}}$ and ${\displaystyle \sigma _{2}}$ are the stresses of constituents 1 and 2 respectively, and we define the homogenized strain as ${\displaystyle {\overline {\epsilon }}}$. Noting stress to be a force per unit area, a force balance gives that

${\displaystyle \sigma _{\infty }=f\sigma _{1}+\left(1-f\right)\sigma _{2}}$

2

Equations 1 and 2 can be combined to give

${\displaystyle {\overline {E}}_{V}{\overline {\epsilon }}=fE_{1}\epsilon _{1}+\left(1-f\right)E_{2}\epsilon _{2}.}$

Finally, since ${\displaystyle {\overline {\epsilon }}=\epsilon _{1}=\epsilon _{2}}$, the overall elastic modulus of the composite can be expressed as ^[6]

${\displaystyle {\overline {E}}_{V}=fE_{1}+\left(1-f\right)E_{2}.}$

Assuming the Poisson's ratio of the two materials is the same, this represents the upper bound of the composite's elastic modulus. ^[7]

Inverse rule of mixtures / Reuss model / equal stress

Alternatively, we can assume that the stress in the two constituents is equal, i.e. ${\displaystyle \sigma _{\infty }=\sigma _{1}=\sigma _{2}}$. This corresponds to the two constituents being loaded in series, which in the context of fiber-reinforced composites roughly corresponds to transverse loading. In this case, the overall strain is distributed according to

${\displaystyle {\overline {\epsilon }}=f\epsilon _{1}+\left(1-f\right)\epsilon _{2}.}$

The overall modulus in the material is then given by

${\displaystyle {\overline {E}}_{R}={\frac {\sigma _{\infty }}{\overline {\epsilon }}}={\frac {\sigma _{1}}{f\epsilon _{1}+\left(1-f\right)\epsilon _{2}}}=\left({\frac {f}{E_{1}}}+{\frac {1-f}{E_{2}}}\right)^{-1}}$

since ${\displaystyle \sigma _{1}=E_{1}\epsilon _{1}}$, ${\displaystyle \sigma _{2}=E_{2}\epsilon _{2}}$. ^[6]

Other properties

Similar derivations give the rules of mixtures for

• mass density:$${\displaystyle \left({\frac {f}{\rho _{1}}}+{\frac {1-f}{\rho _{2}}}\right)^{-1}\leq \ {\overline {\rho }}\leq f\rho _{1}+(1-f)\rho _{2}}$$

• thermal conductivity:$${\displaystyle \left({\frac {1}{k_{1}}}+{\frac {1-f}{k_{2}}}\right)^{-1}\leq {\overline {k}}\leq fk_{1}+\left(1-f\right)k_{2}}$$

• electrical conductivity:$${\displaystyle \left({\frac {f}{\sigma _{1}}}+{\frac {1-f}{\sigma _{2}}}\right)^{-1}\leq {\overline {\sigma }}\leq f\sigma _{1}+\left(1-f\right)\sigma _{2}}$$

Notably, these forms are not applicable to strength-related properties. Consider the Reuss assumption, that the stress is equal in both constituents. In this case, once the material is loaded to the ultimate tensile strength of the weakest constituent, that material will fail, which immediately implies failure of the composite. Thus, the ultimate tensile strength of the composite would be that of the weakest phase. Note that this in turn assumes there are no other failure mechanisms, such as debonding, which get activated before either constituent fails.

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: ^[8]

${\displaystyle ({\overline {E}})^{k}=f(E_{1})^{k}+(1-f)(E_{2})^{k}}$

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 ${\displaystyle E_{i}}$ and volume fraction ${\displaystyle V_{i}}$, where

${\displaystyle \sum _{i=1}^{n}{V_{i}}=1,}$

then the rule of mixtures can be shown to give:

${\displaystyle {\overline {E}}=\sum _{i=1}^{n}{V_{i}E_{i}}}$

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

${\displaystyle {\frac {1}{\overline {E}}}=\sum _{i=1}^{n}{\frac {V_{i}}{E_{i}}}.}$

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

${\displaystyle ({\overline {E}})^{k}=\sum _{i=1}^{n}V_{i}(E_{i})^{k}}$

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

References

1. Alger, Mark. S. M. (1997). Polymer Science Dictionary (2nd ed.). Springer Publishing. ISBN   0412608707.
2. "Stiffness of long fibre composites". University of Cambridge . Retrieved 1 January 2013.
3. Askeland, Donald R.; Fulay, Pradeep P.; Wright, Wendelin J. (2010-06-21). The Science and Engineering of Materials (6th ed.). Cengage Learning. ISBN   9780495296027.
4. 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.
5. 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.
6. "Derivation of the rule of mixtures and inverse rule of mixtures". University of Cambridge . Retrieved 1 January 2013.
7. Yu, Wenbin (2024). "Common Misconceptions on Rules of Mixtures for Predicting Elastic Properties of Composites". AIAA Journal. 62 (5): 1982–1987. Bibcode:2024AIAAJ..62.1982Y. doi:10.2514/1.J063863.
8. Soboyejo, W. O. (2003). "9.3.1 Constant-Strain and Constant-Stress Rules of Mixtures". Mechanical properties of engineered materials. Marcel Dekker. ISBN   0-8247-8900-8. OCLC   300921090.

External links

• Rule of mixtures calculator