Two previously undeformed specimens of the same metal are to be plastically deformed by reducing their cross-sectional areas. One has a circular cross section, and the other is rectangular; during deformation the circular cross section is to remain circular, and the rectangular is to remain as such. Their original and deformed dimensions are as follows: $$ \begin{array}{lcr} \hline & \begin{array}{c} \text { Circular } \\ \text { (diameter, } \mathbf{m m}) \end{array} & \begin{array}{c} \text { Rectangular } \\ (\mathbf{m m}) \end{array} \\ \hline \text { Original dimensions } & 15.2 & 125 \times 175 \\ \text { Deformed dimensions } & 11.4 & 75 \times 200 \\ \hline \end{array} $$ Which of these specimens will be the hardest after plastic deformation, and why?

Step by step solution

01

Calculate the initial and final areas of circular and rectangular specimens

First, we need to calculate the initial and final cross-sectional areas for both the circular and rectangular specimens. For the circular specimen, we have: Initial diameter = 15.2 mm Final diameter = 11.4 mm So, initial area of the circular specimen is given by: $$A_{ci} = \pi(\frac{D_{c_i}}{2})^2 = \pi(\frac{15.2}{2})^2$$ And the final area of the circular specimen is given by: $$A_{cf} = \pi(\frac{D_{c_f}}{2})^2 = \pi(\frac{11.4}{2})^2$$ For the rectangular specimen, we have: Initial dimensions = 125 mm × 175 mm Final dimensions = 75 mm × 200 mm So, the initial area of the rectangular specimen is given by: $$A_{ri} = 125 \times 175$$ And the final area of the rectangular specimen is given by: $$A_{rf} = 75 \times 200$$

02

Calculate the percentage change in area for both specimens

Now, we calculate the percentage change in the area of both specimens as follows: For the circular specimen: $$\% \text{ change}_c = \frac{A_{ci}-A_{cf}}{A_{ci}} \times 100$$ For the rectangular specimen: $$\% \text{ change}_r = \frac{A_{ri}-A_{rf}}{A_{ri}} \times 100$$

03

Compare the percentage change in area and determine the hardest specimen

After calculating the percentage change in the area of both specimens, compare their values to determine which specimen has a higher percentage change. The specimen with the higher percentage change in the area will be the hardest after plastic deformation as a higher percentage change in the area is directly related to more significant work hardening and greater hardness. In conclusion, by comparing the percentage change in the area of both circular and rectangular specimens, we can determine which one of them will be the hardest after the plastic deformation and also provide an explanation for the observed difference in their hardness.

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