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 .[1][2][3] It provides a theoretical upper- and lower-bound on properties such as the elastic modulus, ultimate tensile strength, thermal conductivity, and electrical conductivity.[3] In general there are two models, one for axial loading (Voigt model),[2][4] and one for transverse loading (Reuss model).[2][5]
In general, for some material property (often the elastic modulus[1]), the rule of mixtures states that the overall property in the direction parallel to the fibers may be as high as
where
is the volume fraction of the fibers
is the material property of the fibers
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
If the property under study is the elastic modulus, this quantity is called the lower-bound modulus, and corresponds to a transverse loading.[2]
^ abAlger, Mark. S. M. (1997). Polymer Science Dictionary (2nd ed.). Springer Publishing. ISBN 0412608707.
^ abcd"Stiffness of long fibre composites". University of Cambridge. Retrieved 1 January 2013.
^ abAskeland, 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.
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