A new method for the evaluation of the effective properties of composites containing unidirectional periodic nanofibers

Abstract

We propose an innovative numerical scheme (based on complex variable techniques) for the calculation of the effective properties of a composite containing unidirectional periodic fibers in which we additionally incorporate the separate contribution of the ‘interface effect’ between the fibers and the surrounding material. The incorporation of interface effect into the model of deformation allows our model to accommodate the general class of nanocomposite materials, a fast growing area of research and our primary focus in this paper. The composite is loaded by a constant normal strain along the direction parallel to the fibers and by a uniform remote loading in the plane perpendicular to the fibers. Our method is based on the analysis of a representative unit cell with periodic boundary conditions imposed on its edge. Several examples are presented to study the influence of the interface and the volume fraction of the fibers on the effective properties of the composite and the interfacial stress field. We show that when the volume fraction falls below roughly 9%, the interfacial stress distribution recovers effectively to that corresponding to a single fiber with the same interface parameters embedded within an infinite matrix. We find also that if the shear modulus of the fibers exceeds approximately two and a half times that of the matrix, the interface effect is negligible in the determination of the effective properties of the corresponding nanocomposites. Finally, we show that the use of traditional effective medium theories may induce significant errors in the determination of transverse effective properties (in the plane perpendicular to the fibers) of the composite, in particular when the fibers are significantly softer than the matrix.

Appendix

The average out-of-plane normal stress in the RUC is defined by

$$\begin{aligned} \sigma _{33}^{\mathrm{ave}} =\frac{\iint _{S_0 } {\sigma _{33}^{(0)} \text {d}x_1 \text {d}x_2 } +\iint _{S_1 } {\sigma _{33}^{(1)} \text {d}x_1 \text {d}x_2 } +\oint _L {\sigma _{33}^{(S)} \text {d}s} }{a^{2}}. \end{aligned}$$

(43)

In (43), using Green’s theorem and Eqs. (4), (6), (7) and (14), the integrals over the region $S_{1}$ and the curve L can be given explicitly by the coefficient $a_{11}$ as

$$\begin{aligned} \iint _{S_1 } {\sigma _{33}^{(1)} \text {d}x_1 \text {d}x_2 }= & {} 2\nu _1 \hbox {Im}\left[ {\oint _L {\overline{\varphi _1 (z)} \text {d}z} } \right] +2\pi R^{2}(1+\nu _1 )G_1 \varepsilon _{33} \nonumber \\= & {} 4\pi R\nu _1 \hbox {Re}(a_{11} )+2\pi R(1+\nu _1 )\Delta _0 {G_1 }/{G_0 }, \end{aligned}$$

(44)

$$\begin{aligned} \oint _L {\sigma _{33}^{(S)} \text {d}s}= & {} 2\pi R\gamma \Delta _0 +\eta G_0 R\oint _L {\varepsilon _{tt} \text {d}s} \nonumber \\= & {} 2\pi R\left[ {(\gamma -\nu _1 \eta )\Delta _0 +\frac{\eta (1-2\nu _1 )\hbox {Re}(a_{11} )}{{G_1 }/{G_0 }}} \right] , \end{aligned}$$

(45)

while the integral over the region $S_{0}$ can be evaluated from

$$\begin{aligned} \iint _{S_0 } {\sigma _{33}^{(0)}\text {d}x_1 \text {d}x_2 }= & {} 2\nu _0 \hbox {Im}\left[ {\oint _{DBAC} {\overline{\varphi _0 (z)} \text {d}z} -\oint _L {\overline{\varphi _0 (z)} \text {d}z} } \right] \nonumber \\&\quad +\,2(1+\nu _0 )\Delta _0 \left( {{a^{2}}/R-\pi R} \right) . \end{aligned}$$

(46)

via numerical quadrature.