Chapter 12.15, Problem 86AAP

(a)

To determine

The first crack is formed in the composite.

Answer to Problem 86AAP

The first crack is formed in the composite is on ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}\text{matrix}$.

Explanation of Solution

Write the expression to calculate strain on the composite $\left({\epsilon }_{\text{comp}}\right)$.

 ${\epsilon }_{\text{comp}}=\frac{{\sigma }_{\text{comp}}}{E_{\text{comp}}}$

Here, stress on the composite is ${\sigma }_{\text{comp}}$ and modulus of elasticity on the composite is $E_{\text{comp}}$.

Write the fracture-toughness equation for the composite.

 $K_{\text{IC,}\text{comp}}=Y{\sigma }_{\text{comp}}\sqrt{\pi a}$

Here, fracture-toughness of the composite is $K_{\text{IC,}\text{comp}}$, dimensionless constant is $Y$ and lattice constant is $a$.

For the isostrain conditions, the strain on the composite will be equal to strain on the matrix and strain on the fiber respectively.

 $\begin{array}{l}{\epsilon }_{\text{comp}}={\epsilon }_{\text{SiC}}={\epsilon }_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}\\ \frac{{\sigma }_{\text{comp}}}{E_{\text{comp}}}=\frac{{\sigma }_{\text{SiC}}}{E_{\text{SiC}}}=\frac{{\sigma }_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}}{E_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}}\end{array}$

Here, strain on the $\text{SiC}$ fiber is ${\epsilon }_{\text{SiC}}$, strain on the ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ matrix is ${\epsilon }_{\text{RBSN}}$, stress on the $\text{SiC}$ fiber is ${\sigma }_{\text{SiC}}$, modulus of elasticity on the $\text{SiC}$ fiber is $E_{\text{SiC}}$, stress on the ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ matrix is ${\sigma }_{\text{RBSN}}$ and modulus of elasticity on the ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ matrix is $E_{\text{RBSN}}$.

Considering the $\text{SiC}$ fibers, write the expression to calculate the stress on the composite.

 $\frac{{\sigma }_{\text{comp}}}{E_{\text{comp}}}=\frac{{\sigma }_{\text{SiC}}}{E_{\text{SiC}}}$

 $\begin{array}{c}{\sigma }_{\text{comp}}=\frac{E_{\text{comp}}}{E_{\text{SiC}}}\left({\sigma }_{\text{SiC}}\right)\\ =\frac{E_{\text{comp}}}{E_{\text{SiC}}}\left(\frac{K_{\text{IC,}\text{SiC}}}{Y\sqrt{\pi a}}\right)\end{array}$                                                                                             (I)

Here, fracture-toughness of the $\text{SiC}$ fiber is $K_{\text{IC,}\text{SiC}}$ and surface notch on the $\text{SiC}$ fiber is $a$.

Considering the ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ matrix, write the expression to calculate the stress on the composite.

 $\frac{{\sigma }_{\text{comp}}}{E_{\text{comp}}}=\frac{{\sigma }_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}}{E_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}}$

 $\begin{array}{c}{\sigma }_{\text{comp}}=\frac{E_{\text{comp}}}{E_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}}\left({\sigma }_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}\right)\\ =\frac{E_{\text{comp}}}{E_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}}\left(\frac{K_{\text{IC,}{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}}{Y\sqrt{\pi a}}\right)\end{array}$                                                                                        (II)

Here, fracture-toughness of the ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ matrix is $K_{\text{IC,}{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$ and surface flaw on the ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ matrix is $a$.

Conclusion:

Substitute $347\text{GPa}$ for $E_{\text{comp}}$, $340\text{GPa}$ for $E_{\text{SiC}}$, $4.6\text{MPa}\sqrt{\text{m}}$ for $K_{\text{IC,}\text{SiC}}$, $1$ for $Y$ and $4.5\times {10}^{-6}\text{m}$ for $a$ in Equation (I).

 $\begin{array}{c}{\sigma }_{\text{comp}}=\frac{347\text{GPa}}{340\text{GPa}}\left(\frac{4.6\text{MPa}\sqrt{\text{m}}}{\left(1\right)\sqrt{\pi \left(4.5\times {10}^{-6}\text{m}\right)}}\right)\\ =1249\text{MPa}\end{array}$

Substitute $347\text{GPa}$ for $E_{\text{comp}}$, $350\text{GPa}$ for $E_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$, $3.8\text{MPa}\sqrt{\text{m}}$ for $K_{\text{IC,}{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$, $1$ for $Y$ and $5.0\times {10}^{-6}\text{m}$ for $a$ in Equation (II).

 $\begin{array}{c}{\sigma }_{\text{comp}}=\frac{347\text{GPa}}{350\text{GPa}}\left(\frac{3.8\text{MPa}\sqrt{\text{m}}}{\left(1\right)\sqrt{\pi 5\left(3.0\times {10}^{-6}\text{m}\right)}}\right)\\ =951\text{MPa}\end{array}$

When compared to $\text{SiC}$ fiber, the ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ matrix will start to crack first at the applied stress of $951\text{MPa}$.

Thus, the first crack is formed in the composite is on ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}\text{matrix}$.

(b)

To determine

The stress at which first crack is formed in the composite.

Answer to Problem 86AAP

The stress at which first crack is formed in the composite is $\text{951}\text{MPa}$.

Explanation of Solution

Conclusion:

When compared to $\text{SiC}$ fiber, the ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ matrix will start to crack first at the applied stress of $951\text{MPa}$.

Thus, the stress at which first crack is formed in the composite is $\text{951}\text{MPa}$.