Chapter 12.15, Problem 47AAP

To determine

Derive an equation showing the relationship between the elastic modulus of a layered composite of made of plastic matrix and unidirectional fibers subjected to isostress conditions.

Answer to Problem 47AAP

The equation of elastic modulus of the layered composite is $\frac{E_f{E}_m}{E_m{V}_f+E_f{V}_m}$.

Explanation of Solution

For the isostress conditions, the stress on the composite will be equal to stress on the matrix and stress on the fiber respectively.

 ${\sigma }_c={\sigma }_f={\sigma }_m$                                                                                                           (I)

Here, stress on the composite is ${\sigma }_c$, stress on the matrix is ${\sigma }_m$ and stress on the fiber is ${\sigma }_f$.

Also, the total change in length of the composite will be equal to summation of changes in lengths of the matrix and fiber respectively.

 $\begin{array}{l}\Delta L_c=\Delta L_f+\Delta L_m\\ {\epsilon }_c{l}_c={\epsilon }_f{l}_f+{\epsilon }_m{l}_m\end{array}$                                                                                                    (II)

Here, total change in length of the composite is $\Delta L_c$, total change in length of the matrix is $\Delta L_m$, total change in length of the fiber is $\Delta L_f$, strain on the composite is ${\epsilon }_c$, strain on the matrix is ${\epsilon }_f$, strain on the fiber is ${\epsilon }_m$, original length of the composite is $l_c$, original length of the matrix is $l_m$, original length of the fiber is $l_f$.

Rewrite Equation (II) in terms of volumes.

 $\begin{array}{l}{\epsilon }_c={\epsilon }_f\frac{l_f}{l_c}+{\epsilon }_m\frac{l_m}{l_c}\\ {\epsilon }_c={\epsilon }_f{V}_f+{\epsilon }_m{V}_m\end{array}$                                                                                              (III)

Here, volumes of the fiber and matrix are $V_f$, and $V_m$ respectively.

Rewrite Equation (III) in terms of stress and modulus using Hooke's law.

 $\begin{array}{l}{\epsilon }_c={\epsilon }_f{V}_f+{\epsilon }_m{V}_m\\ \frac{{\sigma }_c}{E_c}=\left(\frac{{\sigma }_f}{E_f}\right)V_f+\left(\frac{{\sigma }_m}{E_m}\right)V_m\\ \frac{\sigma }{E_c}=\left(\frac{\sigma }{E_f}\right)V_f+\left(\frac{\sigma }{E_m}\right)V_m\end{array}$                                                                                     (IV)

Here, modulus of elasticity of the composite, fiber and matrix are $E_c$, $E_f$ and $E_m$ respectively.

Divide Equation (IV) by $\sigma$.

 $\begin{array}{l}\frac{1}{E_c}=\left(\frac{1}{E_f}\right)V_f+\left(\frac{1}{E_m}\right)V_m\\ \frac{1}{E_c}=\frac{V_f}{E_f}+\frac{V_m}{E_m}\\ \frac{1}{E_c}=\frac{E_m{V}_f+E_f{V}_m}{E_f{E}_m}\\ E_c=\frac{E_f{E}_m}{E_m{V}_f+E_f{V}_m}\end{array}$

Thus, the equation of elastic modulus of the layered composite is $\frac{E_f{E}_m}{E_m{V}_f+E_f{V}_m}$.