Chapter 2, Problem 9P

A part made from annealed AISI 1018 steel undergoes a 20 percent cold-work operation.

1. a) Obtain the yield strength and ultimate strength before and after the cold-work operation. Determine the percent increase in each strength.
2. b) Determine the ratios of ultimate strength to yield strength before and after the cold-work operation. What does the result indicate about the change of ductility of the part?

To determine

The yield strength if annealed AISI 1018 steel undergoes a 20 percent cold-work operation.

The ultimate strength if annealed AISI 1018 steel undergoes a 20 percent cold-work operation.

The percentage increase in yield strength.

The percentage increase in ultimate strength.

Answer to Problem 9P

The yield strength if annealed AISI 1018 steel undergoes a 20 percent cold-work operation is $62.529\text{kpsi}$.

The ultimate strength if annealed AISI 1018 steel undergoes a 20 percent cold-work operation $61.875\text{kpsi}$.

The percentage increase in yield strength is $95.40\%$.

The percentage increase in ultimate strength is $25\%$.

Explanation of Solution

Write the expression for work load factor and area.

$\frac{A_0}{A_i}=\frac{1}{1-W}$ (I)

Here, the original area is $A_0$, the area after load is $A_i$ and work load factor is $W$.

Write the expression for true strain.

${\epsilon }_i=\ln \frac{A_0}{A_i}$ (II)

Here, the true strain is ${\epsilon }_i$.

Calculate the new yield strength after cold work.

${S^{\prime }}_y={\sigma }_0{\epsilon }^m{}_i$ (III)

Here, the new yield strength is ${S^{\prime }}_y$, the strength coefficient is ${\sigma }_0$ and strain strengthening exponent is $m$.

Calculate the new ultimate strength after cold work.

${S^{\prime }}_u=\frac{S_u}{1-W}$ (IV)

Here, the new ultimate strength after clod work is ${S^{\prime }}_u$ and the ultimate strength before cold work is $S_u$.

Calculate the percentage increase in yield strength.

${\text{d}}_y\text{=}\left(\frac{\left({S^{\prime }}_y-S_y\right)}{\left(S_y\right)}\right)\times 100\%$                                               (V)

Here, the percentage increase in yield strength is ${\text{d}}_y$, the yield strength before cold-work is $S_y$.

Calculate the percentage increase in ultimate strength.

${\text{d}}_u=\left(\frac{\left({S^{\prime }}_u-S_u\right)}{\left(S_u\right)}\right)\times 100\%$ (VI)

Here, the percentage increase in ultimate strength is ${\text{d}}_u$.

Conclusion:

Refer to Table A-22 “Results of Tensile Test of Some Metals” for annealed AISI 1018 steel.

Obtain the yield strength as $32\text{kpsi}$, the ultimate strength before cold work as $49.5\text{kpsi}$, the strength coefficient as $90\text{kpsi}$ and strain strengthening exponent as $0.25$.

The steel undergoes $20$ percent cold work operation therefore the value of work load factor $W$ is $0.20$.

Substitute $0.20$ for $W$ in Equation (I).

$\begin{array}{c}\frac{A_0}{A_i}=\frac{1}{1-0.20}\\ =1.25\end{array}$

Substitute $1.25$ for $\frac{A_0}{A_i}$ in Equation (II).

$\begin{array}{c}{\epsilon }_i=\ln 1.25\\ =0.223\end{array}$

Substitute $0.223$ for ${\epsilon }_i$, $0.25$ for $m$ and $90\text{kpsi}$ for ${\sigma }_0$ in Equation (III).

$\begin{array}{c}{S^{\prime }}_y=\left(90\text{kpsi}\right){\left(0.233\right)}^{0.25}\\ =\left(90\text{kpsi}\right)\left(0.694766\right)\\ \approx 62.529\text{kpsi}\end{array}$

Thus, the yield strength is $62.529\text{kpsi}$.

Substitute $32\text{kpsi}$ for $S_y$ and $62.529\text{kpsi}$ for ${S^{\prime }}_y$ in Equation (V).

$\begin{array}{c}{\text{d}}_y=\left(\frac{\left(62.529\text{kpsi}-32\text{kpsi}\right)}{\left(32\text{kpsi}\right)}\right)\times 100\%\\ =\left(\frac{30.529\text{kpsi}}{32\text{kpsi}}\right)\times 100\%\\ =95.40\%\end{array}$

Thus, the percentage increase in yield strength is $95.40\%$.

Substitute $49.5\text{kpsi}$ for $S_u$ and $0.20$ for $W$ in Equation (IV).

$\begin{array}{c}{S^{\prime }}_u=\frac{\left(49.5\text{kpsi}\right)}{\left(1-0.20\right)}\\ =\frac{\left(49.5\text{kpsi}\right)}{\left(0.80\right)}\\ =61.875\text{kpsi}\end{array}$

Thus, the ultimate strength is $61.875\text{kpsi}$.

Substitute $49.5\text{kpsi}$ for $S_u$ and $61.875\text{kpsi}$ for ${S^{\prime }}_u$ in Equation (VI).

$\begin{array}{c}{\text{d}}_u=\left(\frac{\left(61.875\text{kpsi}\right)-\left(49.5\text{kpsi}\right)}{\left(49.5\text{kpsi}\right)}\right)\times 100\%\\ =\left(\frac{12.375\text{kpsi}}{49.5\text{kpsi}}\right)\times 100\%\\ =25\%\end{array}$

Thus, the percentage increase in ultimate strength is $25\%$.

To determine

The ratio of ultimate and yield strength before cold-work operation.

The ratio of ultimate and yield strength after cold-work operation and explain the ductility of part with the help of result.

Answer to Problem 9P

The ratio of ultimate and yield strength before cold-work operation is $1.55$.

The ratio of ultimate and yield strength after cold-work operation is $1$ and after cold working operation steel lost most of its ductility.

Explanation of Solution

Calculate the ratio of ultimate strength to yield strength before cold work.

$r_1=\frac{S_u}{S_y}$                                                          (VII)

Here, the ratio of ultimate strength to yield strength is $r_1$, the ultimate strength before cold work is $S_u$ and yield strength before cold work is $S_y$.

Calculate the ratio of ultimate strength to yield strength after cold work.

$r_2=\frac{{S^{\prime }}_u}{{S^{\prime }}_y}$ (VIII)

Here, the ratio of ultimate strength to yield strength after cold work is $r_2$, the ultimate strength after cold work is ${S^{\prime }}_u$ and the yield strength after cold work is ${S^{\prime }}_y$.

Conclusion:

Substitute $49.5\text{kpsi}$ for $S_u$ and $32\text{kpsi}$ for $S_y$ in Equation (VII).

$\begin{array}{c}{r}_1=\frac{\left(49.5\text{kpsi}\right)}{\left(32\text{kpsi}\right)}\\ =1.5468\\ \approx 1.55\end{array}$

Thus, the ratio of ultimate strength to yield strength before cold work is $1.55$.

Substitute $61.9\text{kpsi}$ for ${S^{\prime }}_u$ and $61.8\text{kpsi}$ for ${S^{\prime }}_y$ in Equation (VIII).

$\begin{array}{c}{r}_2=\frac{\left(61.9\text{kpsi}\right)}{\left(61.8\text{kpsi}\right)}\\ =1.0016\\ \approx 1\end{array}$

Thus, the ratio of ultimate strength to yield strength after cold work is $1$.

From the calculated value of $r_1$ and $r_2$, it is obtained that $r_1>r_2$. So, it can be concluded that the steel loses its difficulty after the cold working.