Chapter 12.15, Problem 84AAP

To determine

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

Answer to Problem 84AAP

The stress at which first crack is formed in the composite under isostrain condition is on $\text{SiC}\text{fibers}$.

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.

 $\begin{array}{l}{\epsilon }_{\text{comp}}={\epsilon }_{\text{SiC}}={\epsilon }_{\text{RBSN}}\\ \frac{{\sigma }_{\text{comp}}}{E_{\text{comp}}}=\frac{{\sigma }_{\text{SiC}}}{E_{\text{SiC}}}=\frac{{\sigma }_{\text{RBSN}}}{E_{\text{RBSN}}}\end{array}$

Here, strain on the $\text{SiC}$ fiber is ${\epsilon }_{\text{SiC}}$, strain on the $\text{RBSN}$ 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 $\text{RBSN}$ matrix is ${\sigma }_{\text{RBSN}}$ and modulus of elasticity on the $\text{RBSN}$ 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 $\text{RBSN}$ matrix, write the expression to calculate the stress on the composite.

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

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

Here, fracture-toughness of the $\text{RBSN}$ matrix is $K_{\text{IC,}\text{RBSN}}$ and surface flaw on the $\text{RBSN}$ matrix is $a$.

Conclusion:

Substitute $250\text{GPa}$ for $E_{\text{comp}}$, $395\text{GPa}$ for $E_{\text{SiC}}$, $4.8\text{MPa}\sqrt{\text{m}}$ for $K_{\text{IC,}\text{SiC}}$, $1$ for $Y$ and $3.5\times {10}^{-6}\text{m}$ for $a$ in Equation (I).

 $\begin{array}{c}{\sigma }_{\text{comp}}=\frac{250\text{GPa}}{395\text{GPa}}\left(\frac{4.8\text{MPa}\sqrt{\text{m}}}{\left(1\right)\sqrt{\pi \left(3.5\times {10}^{-6}\text{m}\right)}}\right)\\ =916\text{MPa}\end{array}$

Substitute $250\text{GPa}$ for $E_{\text{comp}}$, $155\text{GPa}$ for $E_{\text{RBSN}}$, $3.5\text{MPa}\sqrt{\text{m}}$ for $K_{\text{IC,}\text{RBSN}}$, $1$ for $Y$ and $3.0\times {10}^{-6}\text{m}$ for $a$ in Equation (II).

 $\begin{array}{c}{\sigma }_{\text{comp}}=\frac{250\text{GPa}}{155\text{GPa}}\left(\frac{3.5\text{MPa}\sqrt{\text{m}}}{\left(1\right)\sqrt{\pi \left(3.0\times {10}^{-6}\text{m}\right)}}\right)\\ =1839\text{MPa}\end{array}$

When compared to $\text{RBSN}$ matrix, the $\text{SiC}$ fiber will start to crack first at the applied stress of $916\text{MPa}$.

Thus, the stress at which first crack is formed in the composite under isostrain condition is on $\text{SiC}\text{fibers}$.