Chapter 12.15, Problem 42AAP

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 iso-strain conditions.

Answer to Problem 42AAP

The equation of elastic modulus of the layered composite is $E_c=E_f{V}_f+E_m{V}_m$.

Explanation of Solution

Write the expression to calculate the load on a material.

 $P=\sigma A$                                                                                                                     (I)

Here, stress on the material is $\sigma$, and area of the material is $A$.

Write the expression to calculate the area of the material.

 $A=Vl$                                                                                                                    (II)

Here, length of the material is $l$, and volume of the material is $V$.

Write the expression to calculate the elastic modulus of the material.

 $E=\frac{\sigma }{\epsilon }$                                                                                                                   (III)

Here, strain on the material is $\epsilon$.

Conclusion:

Write the equation for load on the layered composite.

 $P_c=P_f+P_m$                                                                                                           (IV)

Here, load on the fiber layers is $P_f$ and load on the matrix layers is $P_m$.

Using Equation (I), rewrite Equation (IV) in terms of stresses and areas.

 ${\sigma }_c{A}_c={\sigma }_f{A}_f+{\sigma }_m{A}_m$                                                                                               (V)

Here, stresses on the composites, fiber layers and matrix layers are ${\sigma }_c$, ${\sigma }_f$, and ${\sigma }_m$ respectively and areas of the composites, fiber layers and matrix layers are $A_c$, $A_f$, and $A_m$ respectively.

Since the length of the matrix layers and length of the fiber layers are same, the areas can be replaced by their respective volumes.

Using Equation (II), rewrite Equation (V) in terms of volumes and lengths.

 $\begin{array}{l}{\sigma }_c\left(V_{c}l\right)={\sigma }_f\left(V_{f}l\right)+{\sigma }_m\left(V_{m}l\right)\\ {\sigma }_c{V}_c={\sigma }_f{V}_f+{\sigma }_m{V}_m\end{array}$                                                                                (VI)

Here, fractional volumes of the composites, fiber layers and matrix layers are $V_c$, $V_f$, and $V_m$ respectively.

For iso-strain conditions, all strains will be equal. Therefore, divide each term in Equation (VI) by its corresponding strain.

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

Here, elastic modulus of the composites, fiber layers and matrix layers are $E_c$, $E_f$, and $E_m$ respectively.

The fractional volume is generally considered to be unity.

Substitute 1 for $V_c$ in Equation (VII).

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

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