Rule of mixtures

The upper and lower bounds on the elastic modulus of a composite material with the lower bound predicted by the rule of mixtures given here but the upper bound larger than the rule of mixtures given here.^[1] The actual elastic modulus lies between the curves.

In materials science, a general rule of mixtures is a weighted mean used to predict various properties of a composite material .^[2]^[3]^[4] It provides a theoretical upper- and lower-bound on properties such as the elastic modulus, ultimate tensile strength, thermal conductivity, and electrical conductivity.^[4] In general there are two models, one for axial loading (Voigt model),^[3]^[5] and one for transverse loading (Reuss model).^[3]^[6]

In general, for some material property $E$ (often the elastic modulus^[2]), the rule of mixtures states that the overall property in the direction parallel to the fibers may be as high as

E_{c}=fE_{f}+\left(1-f\right)E_{m}

where

$f={\frac {V_{f}}{V_{f}+V_{m}}}$ is the volume fraction of the fibers
$E_{f}$ is the material property of the fibers
$E_{m}$ is the material property of the matrix

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 $E_{c}$ given by this formula. Even if both constituents are isotropic, the real upper bound is $E_{c}$ plus a term in the order of square of the difference of the Poisson's ratios of the two constituents.^[1]

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

E_{c}=\left({\frac {f}{E_{f}}}+{\frac {1-f}{E_{m}}}\right)^{-1}.

If the property under study is the elastic modulus, this quantity is called the lower-bound modulus, and corresponds to a transverse loading.^[3]

Derivation for elastic modulus

Voigt Modulus

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

{\frac {\sigma _{f}}{E_{f}}}=\epsilon _{f}=\epsilon _{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 _{\infty }=f\sigma _{f}+\left(1-f\right)\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 $\sigma _{\infty }=E_{c}\epsilon _{c}$ for some elastic modulus of the composite $E_{c}$ and some strain of the composite $\epsilon _{c}$ , then equations 1 and 2 can be combined to give

E_{c}\epsilon _{c}=fE_{f}\epsilon _{f}+\left(1-f\right)E_{m}\epsilon _{m}.

Finally, since $\epsilon _{c}=\epsilon _{f}=\epsilon _{m}$ , the overall elastic modulus of the composite can be expressed as^[7]

E_{c}=fE_{f}+\left(1-f\right)E_{m}.

Reuss modulus

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

\epsilon _{c}=f\epsilon _{f}+\left(1-f\right)\epsilon _{m}.

The overall modulus in the material is then given by

E_{c}={\frac {\sigma _{\infty }}{\epsilon _{c}}}={\frac {\sigma _{f}}{f\epsilon _{f}+\left(1-f\right)\epsilon _{m}}}=\left({\frac {f}{E_{f}}}+{\frac {1-f}{E_{m}}}\right)^{-1}

since $\sigma _{f}=E\epsilon _{f}$ , $\sigma _{m}=E\epsilon _{m}$ .^[7]

Other properties

Similar derivations give the rules of mixtures for

mass density: $\rho _{c}=\rho _{f}\centerdot f+\rho _{M}\centerdot (1-f)$ where f is the atomic percent of fiber in the mixture.
ultimate tensile strength: $\left({\frac {f}{\sigma _{UTS,f}}}+{\frac {1-f}{\sigma _{UTS,m}}}\right)^{-1}\leq \sigma _{UTS,c}\leq f\sigma _{UTS,f}+\left(1-f\right)\sigma _{UTS,m}$
thermal conductivity: $\left({\frac {f}{k_{f}}}+{\frac {1-f}{k_{m}}}\right)^{-1}\leq k_{c}\leq fk_{f}+\left(1-f\right)k_{m}$
electrical conductivity: $\left({\frac {f}{\sigma _{f}}}+{\frac {1-f}{\sigma _{m}}}\right)^{-1}\leq \sigma _{c}\leq f\sigma _{f}+\left(1-f\right)\sigma _{m}$

References

1 2 Yu, Wenbin (2016). An Introduction to Micromechanics. Switzerland: Trans Tech Publications. pp. 3–24. ISBN 9783038357469.
1 2 Alger, Mark. S. M. (1997). Polymer Science Dictionary (2nd ed.). Springer Publishing. ISBN 0412608707.
1 2 3 4 "Stiffness of long fibre composites". University of Cambridge. Retrieved 1 January 2013.
1 2 Askeland, Donald R.; Fulay, Pradeep P.; Wright, Wendelin J. (2010-06-21). The Science and Engineering of Materials (6th ed.). Cengage Learning. ISBN 9780495296027.
↑ 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.
↑ 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.
1 2 "Derivation of the rule of mixtures and inverse rule of mixtures". University of Cambridge. Retrieved 1 January 2013.

External links

Rule of mixtures calculator

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[Micromechanics-1] 1 2 Yu, Wenbin (2016). An Introduction to Micromechanics. Switzerland: Trans Tech Publications. pp. 3–24. ISBN 9783038357469.

[PSD-2] 1 2 Alger, Mark. S. M. (1997). Polymer Science Dictionary (2nd ed.). Springer Publishing. ISBN 0412608707.

[UoC-3] 1 2 3 4 "Stiffness of long fibre composites". University of Cambridge. Retrieved 1 January 2013.

[SEM-4] 1 2 Askeland, Donald R.; Fulay, Pradeep P.; Wright, Wendelin J. (2010-06-21). The Science and Engineering of Materials (6th ed.). Cengage Learning. ISBN 9780495296027.

[Voigt-5] 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.

[Reuss-6] 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.

[UoCderiv-7] 1 2 "Derivation of the rule of mixtures and inverse rule of mixtures". University of Cambridge. Retrieved 1 January 2013.

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