Chapter 12.15, Problem 97SEP

(a)

To determine

The volume fractions for the fiber and matrix in the laminate.

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation for the rule of mixture for the composite material.

  $E_c=E_f{V}_f+E_m{V}_m$                                                                                                    (I)

Here, modulus of elasticity of composite is $E_c$, modulus of elasticity of fiber is $E_f$, volume fraction of fiber is $V_f$, modulus of elasticity of matrix is $E_m$ and volume fraction of matrix is $V_m$.

Write the equation to calculate the volume fraction of matrix.

  $V_m=1-V_f$                                                                                                              (II)

Conclusion:

It is given that the component is suggested to be made from unidirectional carbon fiber composite. Therefore, the fiber layer will be carbon and the matrix layer will be epoxy.

Take the modulus of elasticity of carbon fiber to be $33\times {\text{10}}^6\text{psi}$ and the modulus of elasticity of epoxy matrix to be $0.5\times {\text{10}}^6\text{psi}$.

Substitute $\text{10}\times {\text{10}}^6\text{psi}$ for $E_c$, $33\times {\text{10}}^6\text{psi}$ for $E_f$, $0.5\times {\text{10}}^6\text{psi}$ for $E_m$ and $\left(1-V_f\right)$ for $V_m$ in Equation (I).

 $\begin{array}{l}\text{10}\times {\text{10}}^6\text{psi}=\left(33\times {\text{10}}^6\text{psi}\right)V_f+\left(0.5\times {\text{10}}^6\text{psi}\right)\left(1-V_f\right)\\ \text{10}\times {\text{10}}^6\text{psi}=\left(33\times {\text{10}}^6\text{psi}\right)V_f+\left(0.5\times {\text{10}}^6\text{psi}\right)-\left(0.5\times {\text{10}}^6\text{psi}\right)V_f\\ \text{10}\times {\text{10}}^6\text{psi}-0.5\times {\text{10}}^6\text{psi}=\left(33\times {\text{10}}^6\text{psi}-0.5\times {\text{10}}^6\text{psi}\right)V_f\\ 9.5\times {\text{10}}^6\text{psi}=\left(32.5\times {\text{10}}^6\text{psi}\right)V_f\end{array}$

 $\begin{array}{c}{V}_f=\frac{9.5\times {\text{10}}^6\text{psi}}{32.5\times {\text{10}}^6\text{psi}}\\ =0.29\end{array}$

Substitute $\text{0}\text{.29}$ for $V_f$ in Equation (II).

 $\begin{array}{c}{V}_m=1-0.29\\ =0.71\end{array}$

Thus, the volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

(b)

To determine

The volume fractions for the fiber and matrix in the laminate.

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation to calculate the percentage in weight reduction.

  $\%\text{Wt}=\frac{{\rho }_{\text{Al}}-{\rho }_{\text{C}}}{{\rho }_{\text{Al}}}\times 100$                                                                                          (III)

Here, density of carbon composite is ${\rho }_{\text{C}}$ and density of aluminum composite is ${\rho }_{\text{Al}}$.

Conclusion:

Refer Appendix I, "Important properties of the selected engineering materials" to obtain the densities of aluminum composite and carbon composite.

 $\begin{array}{l}{\rho }_{\text{Al}}=0.0975\text{lb}/\text{in}{\text{.}}^3\\ {\rho }_{\text{C}}=0.0553\text{lb}/\text{in}{\text{.}}^3\end{array}$.

Substitute $0.0553\text{lb}/\text{in}{\text{.}}^3$ for ${\rho }_{\text{C}}$ and $0.0975\text{lb}/\text{in}{\text{.}}^3$ for ${\rho }_{\text{Al}}$ in Equation (III).

 $\begin{array}{c}\%\text{Wt}=\frac{0.0975\text{lb}/\text{in}{\text{.}}^3-0.0553\text{lb}/\text{in}{\text{.}}^3}{0.0975\text{lb}/\text{in}{\text{.}}^3}\times 100\\ =0.43\times 100\\ =43\%\end{array}$

Therefore, the carbon composite will be 43% lighter when compared to aluminum composite.