Micromechanical analysis of fuzzy fiber reinforced composites

Abstract

A novel fuzzy fiber reinforced composite (FFRC) reinforced with zig-zag single-walled carbon nanotubes (CNTs) and carbon fibers is proposed. The distinct constructional feature of this composite is that the uniformly aligned CNTs are radially grown on the surface of carbon fibers. Analytical models based on the mechanics of materials approach and the Mori–Tanaka method are derived to estimate the effective elastic constants of this proposed FFRC. The values of the effective elastic properties of this composite are estimated with and without considering an interphase between the CNT and the polymer matrix. It has been found that the transverse effective properties of this composite are significantly improved due to the radial growing of CNTs on the surface of carbon fiber. The effective properties are also found to be sensitive to the CNT diameter.

Appendix

Equation 30 as shown in the results and discussion section can be derived as follows:

Referring to Fig. 6, the RVE of the FFRC can be considered as an equilateral triangle. Thus the volume (${\mathbf{V}}^{{{\mathbf{FFRC}}}}$) of the RVE of the FFRC is given by

$$ {\mathbf{V}}^{{{\mathbf{FFRC}}}} = {\frac{\sqrt 3 }{4}}{\mathbf{D}}^{2} {\mathbf{L}} $$

(31)

where D = 2R. The volume (V ^f) of the carbon fiber is

$$ {\mathbf{V}}^{{\mathbf{f}}} = {\frac{{{\uppi}}}{8}}{\mathbf{d}}^{2} {\mathbf{L}} $$

(32)

where d = 2a. Thus the carbon fiber volume fraction (v _f ) in the FFRC can be expressed as

$$ {\mathbf{v}}_{{\mathbf{f}}} = {\frac{{{\mathbf{V}}^{{\mathbf{f}}} }}{{{\mathbf{V}}^{{{\mathbf{FFRC}}}} }}} = {\frac{{{\uppi}}}{2\sqrt 3 }} {\frac{{{\mathbf{d}}^{2} }}{{{\mathbf{D}}^{2} }}} $$

(33)

Using A (3), the carbon fiber volume fraction (${\bar{\mathbf{v}}}_{\mathbf{f}}$) in the CFF can be derived as

$$ {\bar{\mathbf{v}}}_{{\mathbf{f}}} = {\frac{{{\frac{{{\uppi}}}{8}}{\mathbf{d}}^{2} {\mathbf{L}}}}{{{\frac{{{\uppi}}}{8}}{\mathbf{D}}^{2} {\mathbf{L}}}}} = {\frac{2\sqrt 3 }{{{\uppi}}}}{\mathbf{v}}_{{\mathbf{f}}} $$

(34)

The maximum number $\left( {{\mathbf{N}}_{{{\mathbf{CNT}}}} } \right)_{ \max }$ of radially grown aligned CNTs on the surface of the carbon fiber is given by

$$ \left( {{\mathbf{N}}_{{{\mathbf{CNT}}}} } \right)_{ \max } = {\frac{{{{\uppi}}{\mathbf{dL}}}}{{2\left( {{\mathbf{d}}_{{\mathbf{n}}} + 1.7} \right)^{2} }}} $$

(35)

Therefore the volume (V ^CNT) of the CNTs is

$$ {\mathbf{V}}^{{{\mathbf{CNT}}}} = {\frac{{{\uppi}}}{4}}{\mathbf{d}}_{{\mathbf{n}}}^{2} \left( {{\mathbf{R}} - {\mathbf{a}}} \right)\left( {{\mathbf{N}}_{{{\mathbf{CNT}}}} } \right)_{ \max } $$

(36)

Thus the maximum volume fraction $\left( {{\mathbf{v}}_{{{\mathbf{CNT}}}} } \right)_{ \max }$ of the CNT with respect to the volume of the FFRC can be determined as

$$ \left( {{\mathbf{v}}_{{{\mathbf{CNT}}}} } \right)_{ \max } = {\frac{{{\mathbf{V}}^{{{\mathbf{CNT}}}} }}{{{\mathbf{V}}^{{{\mathbf{FFRC}}}} }}} = {\frac{{{{\uppi}}{\mathbf{d}}_{{\mathbf{n}}}^{2} }}{{2\left( {{\mathbf{d}}_{{\mathbf{n}}} + 1.7} \right)^{2} }}}\left( {\sqrt {{\frac{{{{\uppi}}{\mathbf{v}}_{{\mathbf{f}}} }}{2\sqrt 3 }}} - {\mathbf{v}}_{{\mathbf{f}}} } \right) $$

(37)

Finally, the maximum volume fraction (max (v _nt )) of the CNTs with respect to the volume of the PMNC and with respect to the volume of the CFF (max (${\bar{\mathbf{v}}}_{\mathbf{nt}}$)) can be determined in terms of max (v _CNT ) as follows:

$$ ({\mathbf{v}}_{{{\mathbf{nt}}}} )_{ \max } = {\frac{{{\mathbf{V}}^{{{\mathbf{CNT}}}} }}{{{\mathbf{V}}^{{{\mathbf{PMNC}}}} }}} = {\frac{2\sqrt 3 }{{{\uppi}}}}{\frac{{\left( {{\mathbf{D}}^{2} } \right)}}{{\left( {{\mathbf{D}}^{2} - {\mathbf{d}}^{2} } \right)}}}\left( {{\mathbf{v}}_{{{\mathbf{CNT}}}} } \right)_{ \max } $$

(38)

$$ ({\bar{\mathbf{v}}}_{{{\mathbf{nt}}}} )_{ \max } = {\frac{{{\mathbf{V}}^{{{\mathbf{CNT}}}} }}{{{\mathbf{V}}^{{{\mathbf{CFF}}}} }}} = {\frac{2\sqrt 3 }{{{\uppi}}}}\left( {{\mathbf{v}}_{{{\mathbf{CNT}}}} } \right)_{ \max } $$

(39)