Chapter 7, Problem 7.6P

(a)

To determine

All of the constants in the equation fir tensile yield strength as function of solid solution composition.

Answer to Problem 7.6P

The constants in the equation fir tensile yield strength as function of solid solution composition is $8\times {10}^9{\text{ N/m}}^2\cdot {\text{af}}^{1/2}$ .

Explanation of Solution

Concept used:

Draw the diagram for yield strength of iron as a function of the atom fraction of carbon to the one-half power as shown below:

  

Write the expression for tensile strength of solid solution strengthened alloy.

  ${\sigma }_y={\sigma }_0+k_{st}{c}_s^{1/2}$ …… (1)

Here, ${\sigma }_y$ is the tensile strength of solid solution strengthened alloy, ${\sigma }_0$ is the tensile yield stress at zero concentration of solid-solution atoms, $c_s$ is the solute chemical composition in terms of atom fraction and $k_{st}$ is constant.

Calculation:

Refer to figure 1; the value of ${\sigma }_0$ at zero concentration of atom fraction of carbon is $7\times {10}^7{\text{ N/m}}^2$ .

Refer to figure 1; at atom fraction of carbon as $10\times {10}^{-3}{\text{ af}}^{1/2}$ the corresponding yield stress is $15\times {10}^7{\text{ N/m}}^2$ .

Substitute $15\times {10}^7{\text{ N/m}}^2$ for ${\sigma }_y$ , $7\times {10}^7{\text{ N/m}}^2$ for ${\sigma }_0$ and $10\times {10}^{-3}{\text{ af}}^{1/2}$ for $c_s^{1/2}$ in equation (1).

  $\begin{array}{c}15\times {10}^7{\text{ N/m}}^2=7\times {10}^7{\text{ N/m}}^2+k_{st}\left(10\times {10}^{-3}{\text{ af}}^{1/2}\right)\\ k_{st}=\frac{8\times {10}^7{\text{N/m}}^2}{\left(10\times {10}^{-3}{\text{ af}}^{1/2}\right)}\\ k_{st}=8\times {10}^9{\text{ N/m}}^2\cdot {\text{af}}^{1/2}\end{array}$

Conclusion:

Thus, the constants in the equation fir tensile yield strength as function of solid solution composition is $8\times {10}^9{\text{ N/m}}^2\cdot {\text{af}}^{1/2}$ .

(b)

To determine

The yield strength of iron-carbon alloy.

Answer to Problem 7.6P

The yield strength of iron-carbon alloyis $33.53\times {10}^7{\text{N/m}}^2$ .

Explanation of Solution

Substitute $8\times {10}^9{\text{ N/m}}^2\cdot {\text{af}}^{1/2}$ for $k_{st}$ , $7\times {10}^7{\text{ N/m}}^2$ for ${\sigma }_0$ and $0.0011$ for $c_s$ in equation (1).

  $\begin{array}{c}{\sigma }_y=7\times {10}^7{\text{ N/m}}^2+\left(8\times {10}^9{\text{ N/m}}^2\cdot {\text{af}}^{1/2}\right){\left(0.0011\text{ af}\right)}^{1/2}\\ =7\times {10}^7{\text{ N/m}}^2+26.53\times {10}^7{\text{ N/m}}^2\\ =33.53\times {10}^7{\text{N/m}}^2\end{array}$

Conclusion:

Thus, the yield strength of iron-carbon alloyis $33.53\times {10}^7{\text{N/m}}^2$ .

(c)

To determine

Comment if fraction of carbon increased thenthe strength to be given by an extension of the line in the figure or not.

Explanation of Solution

If the concentration of solid solution atoms becomes large then the precipitation of atoms starts to form. As mentioned if concentration is increased from $0.0011$ to $0.0015$ af of carbon the precipitates of iron carbides starts to form in alloy.

The strength dependence of precipitates is different than the strength dependence of solid solution. Therefore, it is not possible to give extension of line in figure.