Specifically random dispersion of discontinuous fibres in the polymer matrix
Pls. find below some references which may be useful:
https://www.researchgate.net/profile/Chunfeng_Yue/publication/223560107_Tensile_properties_of_short-glass-fiber-_and_short-carbon-fiber-reinforced_polypropylene_composites/links/541950490cf203f155add847.pdf
https://www.researchgate.net/profile/Ning_Pan4/publication/216568192_THEORETICAL_DETERMINATION_OF_THE_OPTIMAL_FIBER_VOLUME_FRACTION_AND_FIBER-MATRIX_PROPERTY_COMPATIBILITY_OF_SHORT_FIBER_COMPOSITES/links/0fcfd5092f0611e0ae000000.pdf
http://www.ipme.ru/e-journals/MPM/no_21613/MPM216_02_amuthakkannan.pdf
http://deepblue.lib.umich.edu/bitstream/handle/2027.42/34402/12913_ftp.pdf?sequence=1
http://image.sciencenet.cn/olddata/kexue.com.cn/upload/blog/file/2009/6/2009649326946213.pdf
Article Tensile properties of short-glass-fiber- and short-carbon-fi...
Article Theoretical determination of the optimal fiber volume fracti...
Rather than length of fibers, interaction between fillers and matrix is very important. If the binding between filler and matrix is enough, load transfer between fillers and matrix can occur efficiently, increasing the mechanical reinforcement. If in your case, there is a strong binding, increasing the length might increase the reinforcement of nanocompiosite.
Hi Jimmy,
you should look at the "Halpin-Tsai" equation for the elastic properties of short fibers reinforced composites.
This equation is very useful also because it is based on experimental observations.
You may find the equation in any textbook on mechanics of composite materials.
To summarize the answer: the random modulus can be roughly expressed by the following relationship:
Er=3/8* E11+5/8* E22 (1)
Where E11 and E22 are the longitudinal and the transversal moduli of an ideal unidirectional composite with short fiber aligned
in one direction and having the same volume fraction of fibers of your composite.
Eq. 1 above comes from homogeneization procedures.
Similar results can be obtained (for random fiber composites) with lamination theory by assuming the composite with random fibers be equivalent to a quasi-isotropic laminate [0, +45,-45,90]s where the lamina properties are again E11 and E22 (i.e. calculated properly with H-T equation).
Having all the above in mind now you can realize that , in the limit of long fibers , E11 reachs the highest value representing the upper bound for a given volume fraction of fibers. In this case the H-T equation reduces to the familiar rule of mixtures and E11 is as follows:
E11=Em*Vm+Ef*Vf
Vm and Vf being the volume fractions of phases.
Concerning E22 you will not find much difference with the H-T equation if use the following equation
1/E22=(1/Em)*Vm+(1/Ef)*Vf
(whatever the aspect ratio-the length to the diameter ratio- of the fibers)
Concerning the ultimate properties, namely the strength of random short fiber composites, the story is little bit more complicated. However, please, have a look at
R.M.Jones: "Mechanics of Composite materials" ISBN I-56032-712-X .
In general, the failure properties of the composite will increase until the fiber length exceeds the critical length. Stiffness properties (moduli) plateau at a much shorter length. You may find some useful information in my paper "Short Fiber Orientation and its Effects on the Properties of Thermoplastic Composite Materials" posted on this site. For fiber length distribution data, see also "a Method to Evaluate the Effect of Compounding Technology on the Stress Transfer Interface in Short Fiber Reinforced Thermoplastics" in Polymer Engineering and Science, vol. 18, no. 5, April 1978
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