Chapter 8, Problem 8.82P

Interpretation Introduction

Interpretation:

The cold working process required to produce 0.2 in. diameter of copper wire from the original rod of 2 in. diameter needs to be explained. The process needs to be explained with the hot working.

Concept introduction:

Cold working is a metal forming process performed below recrystallization temperature and hot working is done above the recrystallization temperature.

Answer to Problem 8.82P

73.33 % cold working is required to reduce the diameter of the copper rod from 2 inches to 0.26 in. at $543.\text{2 K}$ the annealing temperature, if the cold working is replaced by hot working copper rod diameter from 2 inches to 0.26 inches.

Explanation of Solution

Given Information:

$\begin{array}{l}{\text{Initial diameter of copper rod,d}}_{\text{0}}\text{=2}\text{in}\hfill \\ {\text{Final diameter of copper wire,d}}_{\text{F}}\text{=0}\text{.2}\text{in}\hfill \\ \text{Minimum yield strength,=60000Psi}\hfill \\ \text{Minimum \% elongation,=5\%}\hfill \\ \text{Maximum cold work,=80\%}\hfill \end{array}$

Calculation:

The figure shows the properties of copper versus the percentage of cold work.

  

From the figure, one can obtain the % of cold work for copper corresponding to 60000 psi yield strength and 5 % elongation as follows:

$\%cw=42\%$

The % cold work is given to calculate the final diameter of the wire as follows:

$\%cw=[\frac{d_0{}^2-d_F{}^2}{d_0{}^2}]\times 100$

Here,

$\begin{array}{l}{\text{d}}_{\text{0}}\text{=Initial diameter of rod}\hfill \\ {\text{d}}_{\text{F}}\text{=Final diameter of rod}\hfill \\ \text{\%cw=Percentage cold work}\hfill \\ \hfill \end{array}$

$\begin{array}{l}42=[\frac{d_0{}^2-{0.2}^2}{d_0{}^2}]\times 100\\ d_0=\sqrt{\frac{0.04}{0.58}}\\ d_0=0.263in\end{array}$

To calculate the temperature,

$T_r=0.4T_m$

Here,

$\begin{array}{l}{\text{T}}_{\text{r}}\text{=Recrystallization temperature}\hfill \\ {\text{T}}_{\text{m}}{\text{=Melting temperature=1085}}^{\text{0}}\text{C}\hfill \end{array}$

$\begin{array}{l}{T}_r=0.4\times (1085+273)\\ T_r=543.2\text{ K}\end{array}$

Hence, annealing is done for 0.263 inches at $543.2\text{ K}$ to restore ductility.

To calculate the percentage cold work for cold working to produce 1 in diameter wire from 2-inch diameter rod.

$\begin{array}{l} \%cw=[\frac{d_0{}^2-d_F{}^2}{d_0{}^2}]\times 100\\ =[\frac{2^2-1^2}{2^2}]\times 100\\ =75\%\end{array}$

Hence, 75 % cold work is required to reduce from 2 into 1 in diameter at $543.2\text{ K}$ temperature.

Now, to calculate % of cold work to reduce diameter from 1 into 0.5 in.

$\begin{array}{l} \%cw=[\frac{d_0{}^2-d_F{}^2}{d_0{}^2}]\times 100\\ =[\frac{1^2-{0.5}^2}{1^2}]\times 100\\ =75\%\end{array}$

Hence 75% cold work is needed toreduce diameter from 1 into 0.5 in at $543.2\text{ K}$ temperature.

To calculate % of the cold world to reduce diameter from 0.5 into 0.263 in.

$\begin{array}{l}\%cw=[\frac{d_0{}^2-d_F{}^2}{d_0{}^2}]\times 100\\ =[\frac{{0.5}^2-{0.263}^2}{{0.5}^2}]\times 100\\ =72.33\%\end{array}$

Hence, 72.33 % cold work needed to reduce diameter from 0.5 into 0.263 in at $543.2\text{ K}$

If cold working is replaced by hot working, then % hot work required to produces a diameter of 0.2 in from 2 in diameter rod is,

$\begin{array}{l} \%HW=[\frac{d_0{}^2-d_F{}^2}{d_0{}^2}]\times 100\\ =[\frac{2^2-{0.26}^2}{2^2}]\times 100\\ =98.27\%\end{array}$

Conclusion

Thus, cold working and hot working formulas are same only hot working does not require, further annealing.