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[[File:Composite elastic modulus.svg|thumb|400px|The upper and lower bounds on the elastic modulus of a composite material, as predicted by the rule of mixtures. 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]] made up of continuous and unidirectional fibers.<ref name="PSD">{{cite book|last=Alger|first=Mark. S. M.|title=Polymer Science Dictionary|edition=2nd|year=1997|publisher=[[Springer Publishing]]|isbn=0412608707}}</ref><ref name="UoC">{{cite web|url=http://www.doitpoms.ac.uk/tlplib/fibre_composites/stiffness.php|title=Stiffness of long fibre composites|publisher=[[University of Cambridge]]|accessdate=1 January 2013}}</ref><ref name="SEM">{{cite book|last1=Askeland|first1=Donald R.|last2=Fulay|first2=Pradeep P.|last3=Wright|first3=Wendelin J.|title=The Science and Engineering of Materials|edition=6th|date=2010-06-21|publisher=[[Cengage Learning]]|isbn=9780495296027}}</ref> 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]].<ref name="SEM" /> In general there are two models, one for axial loading (Voigt model),<ref name="UoC" /><ref name=Voigt>{{cite journal|last=Voigt|first=W.|title=Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper|journal=Annalen der Physik|year=1889|volume=274|pages=573–587|doi=10.1002/andp.18892741206|accessdate=13 June 2013|bibcode = 1889AnP...274..573V }}</ref> and one for transverse loading (Reuss model).<ref name="UoC" /><ref name=Reuss>{{cite journal|last=Reuss|first=A.|title=Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle|journal=ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik|year=1929|volume=9|pages=49–58|doi=10.1002/zamm.19290090104|accessdate=13 June 2013}}</ref>

In general, for some material property <math>E</math> (often the elastic modulus<ref name="PSD" />), the rule of mixtures states that the overall property in the direction parallel to the fibers may be as high as
:<math> E_c = fE_f + \left(1-f\right)E_m </math>
where
* <math>f = \frac{V_f}{V_f + V_m}</math> is the [[volume fraction]] of the fibers
* <math>E_f</math> is the material property of the fibers
* <math>E_m</math> is the material property of the matrix
In the case of the elastic modulus, this is known as the '''upper-bound modulus''', and corresponds to loading parallel to the fibers. 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
:<math>E_c = \left(\frac{f}{E_f} + \frac{1-f}{E_m}\right)^{-1}.</math>
If the property under study is the elastic modulus, this quantity is called the '''lower-bound modulus''', and corresponds to a transverse loading.<ref name="UoC" />

== Derivation for elastic modulus ==

=== Upper-bound modulus ===

Consider a composite material under [[Tension (physics)|uniaxial tension]] <math>\sigma_\infty</math>. If the material is to stay intact, the strain of the fibers, <math>\epsilon_f</math> must equal the strain of the matrix, <math>\epsilon_m</math>. [[Hooke's law]] for uniaxial tension hence gives
{{NumBlk|:|<math>\frac{\sigma_f}{E_f} = \epsilon_f = \epsilon_m = \frac{\sigma_m}{E_m}</math>|{{EquationRef|1}}}}
where <math>\sigma_f</math>, <math>E_f</math>, <math>\sigma_m</math>, <math>E_m</math> 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|:|<math>\sigma_\infty = f\sigma_f + \left(1-f\right)\sigma_m</math>|{{EquationRef|2}}}}
where <math>f</math> is the volume fraction of the fibers in the composite (and <math>1-f</math> 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 <math>\sigma_\infty = E_c\epsilon_c</math> for some elastic modulus of the composite <math>E_c</math> and some strain of the composite <math>\epsilon_c</math>, then equations {{EquationNote|1}} and {{EquationNote|2}} can be combined to give
:<math>E_c\epsilon_c = fE_f\epsilon_f + \left(1-f\right)E_m\epsilon_m.</math>
Finally, since <math>\epsilon_c = \epsilon_f = \epsilon_m</math>, the overall elastic modulus of the composite can be expressed as<ref name="UoCderiv">{{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}}</ref>
:<math> E_c = fE_f + \left(1-f\right)E_m.</math>

=== Lower-bound modulus ===

Now let the composite material be loaded perpendicular to the fibers, assuming that <math>\sigma_\infty = \sigma_f = \sigma_m</math>. The overall strain in the composite is distributed between the materials such that
:<math>\epsilon_c = f\epsilon_f + \left(1-f\right)\epsilon_m.</math>
The overall modulus in the material is then given by
:<math>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}</math>
since <math>\sigma_f=E\epsilon_f</math>, <math>\sigma_m=E\epsilon_m</math>.<ref name="UoCderiv" />

== Other properties ==

Similar derivations give the rules of mixtures for

* [[mass density]]
:<math>\left(\frac{f}{\rho_f} + \frac{1-f}{\rho_m}\right)^{-1} \leq \rho_c \leq f\rho_f + \left(1-f\right)\rho_m </math>

* [[ultimate tensile strength]]
:<math>\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} </math>

* [[thermal conductivity]]
:<math>\left(\frac{f}{k_f} + \frac{1-f}{k_m}\right)^{-1} \leq k_c \leq fk_f + \left(1-f\right)k_m </math>

* [[electrical conductivity]]
:<math>\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 </math>

== References ==
{{reflist}}

[[Category:Materials science]]

Deviation of a local ring - Revision history

en>Mark viking: Added wl