A re-formulation of the Mori–Tanaka method for predicting material properties of fiber-reinforced polymers/composites

Abstract

In this investigation, a re-formulation has been developed to investigate the effect of fiber aspect ratio on the effective elastic moduli of fiber-reinforced polymers/composites. The matrix and inclusions are considered as isotropic materials. The five independent elastic constants are derived based on a modified Mori–Tanaka theory. The relationship between composite elastic constants and inclusion aspect ratio is also established. Three types of composites containing unidirectional aligned fiber and two-dimensional and three-dimensional random orientated inclusions are explicitly analyzed. Moreover, three extreme cases involving long fibers, spheres, and thin discs are taken into account. It is found that the longitudinal elastic properties are very sensitive to fiber-like inclusions, whereas the transverse elastic properties are closely related to disc-like inclusions.

Appendix

For a spherical inclusion with a₁ = a₂ = a₃ and fiber aspect ratio λ = 1, the components of the tensor S are simplified to

$$ {\displaystyle \begin{array}{c}{S}_{1111}={S}_{2222}={S}_{3333}=\frac{7-5{\nu}_0}{15\left(1-{\nu}_0\right)}\\ {}{S}_{1122}={S}_{1133}={S}_{2233}=\frac{5{\nu}_0-1}{15\left(1-{\nu}_0\right)}\\ {}{S}_{1212}={S}_{2323}={S}_{3131}=\frac{4-5{\nu}_0}{15\left(1-{\nu}_0\right)}\end{array}} $$

(32)

For a long fiber-shaped inclusion with aspect ratio $\lambda =\frac{a_1}{a_3}\to \infty$, we have

$$ {\displaystyle \begin{array}{l}{S}_{1111}=0\\ {}{S}_{2222}={S}_{3333}=\frac{5-4{\nu}_0}{8\left(1-{\nu}_0\right)}\\ {}{S}_{2222}={S}_{3333}=\frac{5-4{\nu}_0}{8\left(1-{\nu}_0\right)}\\ {}{S}_{2233}={S}_{3322}=\frac{4{\nu}_0-1}{8\left(1-{\nu}_0\right)}\\ {}{S}_{2211}={S}_{3311}=\frac{\nu_0}{2\left(1-{\nu}_0\right)}\\ {}{S}_{1212}={S}_{1313}=\frac{1}{4}\\ {}{S}_{2323}=\frac{3-4{\nu}_0}{8\left(1-{\nu}_0\right)}\end{array}} $$

(33)

For a thin disc with $\lambda =\frac{a_1}{a_3}\to 0$, we get

$$ {\displaystyle \begin{array}{l}{S}_{1111}=1\\ {}{S}_{2222}={S}_{3333}=0\\ {}{S}_{1122}={S}_{1133}\frac{\nu_0}{1-{\nu}_0}\\ {}{S}_{2233}={S}_{3322}=0\\ {}{S}_{1212}={S}_{1313}=\frac{1}{2}\\ {}{S}_{2323}=0\end{array}} $$

(34)

For the spheroidal inclusion $\lambda =\frac{a_1}{a_3}$, the components of the tensor S are simplified to

$$ {\displaystyle \begin{array}{l}{S}_{1111}=\frac{1}{2\left(1-{\nu}_0\right)}\left\{\left(1-2{\nu}_0\right)+\frac{3{\lambda}^2-1}{\lambda^2-1}-\left[1-2{\nu}_0+\frac{3{\lambda}^2}{\lambda^2-1}\right]g\right\}\\ {}{S}_{2222}={S}_{3333}=\frac{3}{8\left(1-{\nu}_0\right)}\frac{\lambda^2}{\lambda^2-1}+\frac{1}{4\left(1-{\nu}_0\right)}\left[1-2{\nu}_0-\frac{9}{4\left({\lambda}^2-1\right)}\right]g\\ {}{S}_{1212}={S}_{1313}=\frac{1}{4\left(1-{\nu}_0\right)}\left\{1-2{\nu}_0-\frac{\lambda^2+1}{\lambda^2-1}-\frac{1}{2}\left[\left(1-2{\nu}_0\right)-\frac{3\left({\lambda}^2+1\right)}{\lambda^2-1}\right]g\right\}\\ {}{S}_{2323}=\frac{1}{4\left(1-{\nu}_0\right)}\left\{\frac{\lambda^2}{2\left({\lambda}^2-1\right)}+\left[1-2{\nu}_0-\frac{3}{4\left({\lambda}^2-1\right)}\right]g\right\}\kern0.1em \\ {}{S}_{1122}={S}_{1133}=-\frac{1}{2\left(1-{\nu}_0\right)}\left[1-2{\nu}_0+\frac{1}{\lambda^2-1}\right]+\frac{1}{2\left(1-{\nu}_0\right)}\left[1-2{\nu}_0+\frac{3}{2\left({\lambda}^2-1\right)}\right]g\\ {}{S}_{2211}={S}_{3311}=-\frac{1}{2\left(1-{\nu}_0\right)}\frac{\lambda^2}{\lambda^2-1}+\frac{1}{4\left(1-{\nu}_0\right)}\left[\frac{3{\lambda}^2}{\lambda^2-1}-\left(1-2{\nu}_0\right)\right]g\\ {}{S}_{2233}={S}_{3322}=\frac{1}{4\left(1-{\nu}_0\right)}\left\{\frac{\lambda^2}{2\left({\lambda}^2-1\right)}-\left[1-2{\nu}_0+\frac{3}{4\left({\lambda}^2-1\right)}\right]g\right\}\kern0.1em \end{array}} $$

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where ν₀ is the Poisson ratio of the matrix, λ is the aspect ratio of the inclusion, and g is given by

$g=\frac{\lambda }{{\left(1-{\lambda}^2\right)}^{3/2}}\left[\operatorname{arccos}\left(\lambda \right)-\lambda {\left(1-{\lambda}^2\right)}^{1/2}\right]$, prolate shape a₁ > a₂ = a₃.

$\mathrm{g}=\frac{\lambda }{{\left({\lambda}^2-1\right)}^{3/2}}\left[\lambda {\left({\lambda}^2-1\right)}^{1/2}-\operatorname{arccos}h\left(\lambda \right)\right]$, oblate shapea₁ < a₂ = a₃.