Micromechanical Modeling of Polymeric Composite Materials with Moisture Absorption

Abstract

Natural fiber reinforced composites and wood cell wall are two typical polymeric composite materials that can uptake large amounts of water in the humid environment. This chapter presents a micromechanical scheme to study the mechanical degradation of these polymeric composites induced by moisture absorption that takes place in the matrix and/or the reinforcing phase. The moisture absorption and the mechanical degradation are two thermodynamic processes correlated with each other. Taking both processes into consideration, a modified Mori–Tanaka method with introduced damage variables is proposed. The moisture absorption and the mechanical degradation are, respectively, described by eigenstrains and damage variables. After specifying this model with different inclusion shapes, the overall swelling deformation and the mechanical degradation of the randomly oriented and the unidirectional straight fiber reinforced polymeric composites are studied. The theoretical predictions are compared with the experimental results from other literature and a good agreement is obtained.

Appendix A

The bulk and the shear moduli of natural fiber and matrix are, respectively, expressed as

$$ \left\{\begin{array}{l}{\kappa}^{(i)}=\frac{E^{(i)}}{3\left(1-2{\upsilon}^{(i)}\right)}\\ {}{\mu}^{(i)}=\frac{E^{(i)}}{2\left(1+{\upsilon}^{(i)}\right)}\end{array}\right.\kern2.25em \left(i=1,2\right) $$

(16.A.1)

In our theoretical predictions, Poison’s ratios are ${\upsilon}^{(1)}={\upsilon}_0^{(1)}=0.25=1/4$ for sisal fiber and ${\upsilon}^{(2)}={\upsilon}_0^{(2)}=0.33\approx 1/3$ for polypropylene matrix. Substituting Eq. (16.A.1) into (16.24), we arrive at

$$ \begin{array}{l}\kappa ={E}^{(2)}+\dfrac{f\left(2{E}^{(1)}/3-{E}^{(2)}\right)\left(3{E}^{(2)}+2{E}^{(1)}/5+9{E}^{(2)}/8\right)}{2\left(1-f\right){E}^{(1)}+3{fE}^{(2)}+9{E}^{(2)}/8+2{E}^{(1)}/5}\\ \\ {}\mu =\dfrac{3{E}^{(2)}}{8}-\dfrac{f\left(2{E}^{(1)}/5-3{E}^{(2)}/8\right)\left(3{E}^{(2)}{\delta}_1/2+2{E}^{(1)}{\delta}_2\right)}{3{E}^{(2)}\left(2{E}^{(1)}/5+3{E}^{(2)}/8\right){\eta}_1-3{E}^{(2)}{\eta}_2/8+2{E}^{(1)}\left({\eta}_3-{\eta}_4\right)}\end{array} $$

(16.A.2)

As seen in Eq. (16.A.2), the bulk and shear moduli of the composite are expressed by Young’s moduli of the fiber and the matrix (i.e., E ⁽¹⁾ and E ⁽²⁾).

By means of Eqs. (16.25) and (16.A.2), we can fit the experimental data of (E ₀, f) (the scatter points in Fig. 16.4) by the commercial software Mathematica 7.0 with the command FindFit to obtain the moduli ${E}_0^{(1)}=15.03\ \mathrm{GPa}$ and ${E}_0^{(2)}=2.37\ \mathrm{GPa}$, as shown in Fig. 16.4. The fitting procedure is based on the least square method. The square of the residuals R between the results of theoretical predictions and experimental data should be minimized such as

$$ \left\{\begin{aligned}&R=\sum_{i=1}^n{\left[E\left({E}_0^{(1)},{E}_0^{(2)},f\right)-{E}_0\right]}^2\\ &\dfrac{\partial R}{\partial {E}_0^{(1)}}=0\\ &\dfrac{\partial R}{\partial {E}_0^{(2)}}=0\end{aligned}\right. $$

(16.A.3)

from which the two material parameters ${E}_0^{(1)}$ and ${E}_0^{(2)}$ are finally determined.