Rule of mixtures

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

{{NumBlk|:|$\backslash frac\{\backslash sigma\_f\}\{E\_f\}\; =\; \backslash epsilon\_f\; =\; \backslash epsilon\_m\; =\; \backslash frac\{\backslash sigma\_m\}\{E\_m\}$|{{EquationRef|1}}}}

where $\backslash sigma\_f$, $E\_f$, $\backslash sigma\_m$, $E\_m$ 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

{{NumBlk|:|$\backslash sigma\_\backslash infty\; =\; f\backslash sigma\_f\; +\; \backslash left(1-f\backslash right)\backslash sigma\_m$|{{EquationRef|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 $\backslash sigma\_\backslash infty\; =\; E\_\backslash parallel\backslash epsilon\_c$ for some elastic modulus of the composite parallel to the fibres $E\_\backslash parallel$ and some strain of the composite $\backslash epsilon\_c$, then equations {{EquationNote|1}} and {{EquationNote|2}} can be combined to give

:$E\_\backslash parallel\backslash epsilon\_c\; =\; fE\_f\backslash epsilon\_f\; +\; \backslash left(1-f\backslash right)E\_m\backslash epsilon\_m.$

Finally, since $\backslash epsilon\_c\; =\; \backslash epsilon\_f\; =\; \backslash epsilon\_m$, the overall elastic modulus of the composite can be expressed as{{cite web|url=http://www.doitpoms.ac.uk/tlplib/bones/derivation_mixture_rules.php|title=Derivation of the rule of mixtures and inverse rule of mixtures|publisher=University of Cambridge|accessdate=1 January 2013}}

:$E\_\backslash parallel\; =\; fE\_f\; +\; \backslash left(1-f\backslash right)E\_m.$

Assuming the Poisson's ratio of the two materials is the same, this represents the upper bound of the composite's elastic modulus.{{cite journal|title = Common Misconceptions on Rules of Mixtures for Predicting Elastic Properties of Composites| journal = AIAA Journal | author = Yu, Wenbin | year = 2024 | volume = 62 | issue = 5 | pages = 1982–1987 | doi = 10.2514/1.J063863| bibcode = 2024AIAAJ..62.1982Y }}

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

:$\backslash epsilon\_c\; =\; f\backslash epsilon\_f\; +\; \backslash left(1-f\backslash right)\backslash epsilon\_m.$

The overall modulus in the material is then given by

:$E\_\backslash perp\; =\; \backslash frac\{\backslash sigma\_\backslash infty\}\{\backslash epsilon\_c\}\; =\; \backslash frac\{\backslash sigma\_f\}\{f\backslash epsilon\_f\; +\; \backslash left(1-f\backslash right)\backslash epsilon\_m\}\; =\; \backslash left(\backslash frac\{f\}\{E\_f\}\; +\; \backslash frac\{1-f\}\{E\_m\}\backslash right)^\{-1\}$

since $\backslash sigma\_f=E\_f\backslash epsilon\_f$, $\backslash sigma\_m=E\_m\backslash epsilon\_m$.

Similar derivations give the rules of mixtures for

• mass density:$$\backslash left(\backslash frac\{f\}\{\backslash rho\_f\}+\backslash frac\{1-f\}\{\backslash rho\_c\}\backslash right)^\{-1\}\; \backslash leq\backslash rho\_f\backslash centerdot\; f+\backslash rho\_M\backslash centerdot\; (1-f)$$ where f is the atomic percent of fiber in the mixture.
• ultimate tensile strength:$$\backslash left(\backslash frac\{f\}\{\backslash sigma\_\{UTS,f\}\}\; +\; \backslash frac\{1-f\}\{\backslash sigma\_\{UTS,m\}\}\backslash right)^\{-1\}\; \backslash leq\; \backslash sigma\_\{UTS,c\}\; \backslash leq\; f\backslash sigma\_\{UTS,f\}\; +\; \backslash left(1-f\backslash right)\backslash sigma\_\{UTS,m\}$$
• thermal conductivity:$$\backslash left(\backslash frac\{f\}\{k\_f\}\; +\; \backslash frac\{1-f\}\{k\_m\}\backslash right)^\{-1\}\; \backslash leq\; k\_c\; \backslash leq\; fk\_f\; +\; \backslash left(1-f\backslash right)k\_m$$
• electrical conductivity:$$\backslash left(\backslash frac\{f\}\{\backslash sigma\_f\}\; +\; \backslash frac\{1-f\}\{\backslash sigma\_m\}\backslash right)^\{-1\}\; \backslash leq\; \backslash sigma\_c\; \backslash leq\; f\backslash sigma\_f\; +\; \backslash left(1-f\backslash right)\backslash sigma\_m$$

A generalized equation for any loading condition between isostrain and isostress can be written as:{{cite book |last=Soboyejo |first=W. O. |title=Mechanical properties of engineered materials |date=2003 |publisher=Marcel Dekker |isbn=0-8247-8900-8 |chapter=9.3.1 Constant-Strain and Constant-Stress Rules of Mixtures |oclc=300921090}}

:$(E\_c)^k\; =\; f(E\_f)^k\; +\; (1-f)(E\_m)^k$

where k is a value between 1 and −1.

For a composite containing a mixture of n different materials, each with a material property $E\_i$ and volume fraction $V\_i$, where

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

then the rule of mixtures can be shown to give:

:$E\_c\; =\; \backslash sum\_\{i\; =\; 1\}^\{n\}\{V\_i\; E\_i\}$

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

:$\backslash frac\{1\}\{E\_c\}\; =\; \backslash sum\_\{i\; =\; 1\}^\{n\}\{\backslash frac\{V\_i\}\{E\_i\}\}\; .$

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

:$(E\_c)^k\; =\; \backslash sum\_\{i=1\}^\{n\}V\_i(E\_i)^k$

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

{{reflist}}

• [http://www.fxsolver.com/solve/share/nn7-FXn1ZV5hI-IDLBT1zw==/ Rule of mixtures calculator]

Category:Materials science

Category:Laws of thermodynamics