The following tabulated data were gathered from a series of Charpy impact tests on a commercial low-carbon steel alloy. $$ \begin{array}{|cc|} \hline \text { Temperature }\left({ }^{\circ} \boldsymbol{C}\right) & \text { Impact Energy (J) } \\ \hline 50 & 76 \\ \hline 40 & 76 \\ \hline 30 & 71 \\ \hline 20 & 58 \\ \hline 10 & 38 \\ \hline 0 & 23 \\ \hline-10 & 14 \\ \hline-20 & 9 \\ \hline-30 & 5 \\ \hline-40 & 1.5 \\ \hline \end{array} $$ (a) Plot the data as impact energy versus temperature. (b) Determine a ductile-to-brittle transition temperature as the temperature corresponding to the average of the maximum and minimum impact energies. (c) Determine a ductile-to-brittle transition temperature as the temperature at which the impact energy is \(20 \mathrm{~J}\).

Short Answer

Expert verified
Answer: The ductile-to-brittle transition temperature corresponding to the average of maximum and minimum impact energies is approximately 25.44 °C. The ductile-to-brittle transition temperature at which the impact energy is 20 J is around 2.8 °C.

Step by step solution

01

Plot the Data

To plot the data, create a coordinate system with "Temperature (°C)" on the x-axis and "Impact Energy (J)" on the y-axis. Then plot the data points given in the table using a suitable scale.
02

Calculate the Average of Maximum and Minimum Impact Energies

In order to find the ductile-to-brittle transition temperature where the metal behaves both ductile and brittle, we will calculate the average of maximum and minimum impact energies. First, find the maximum and minimum impact energies from the given data: Maximum Impact Energy: 76 J Minimum Impact Energy: 1.5 J Next, calculate the average of maximum and minimum impact energies: Average Impact Energy \(\displaystyle =\dfrac{\text{Maximum Impact Energy}+\text{Minimum Impact Energy}}{2}\) \(\displaystyle =\dfrac{76\ \textnormal{J}+1.5\ \textnormal{J}}{2}\) \(\displaystyle = 38.75\ \textnormal{J}\)
03

Find the Ductile-to-Brittle Transition Temperature

Now we need to find out the ductile-to-brittle transition temperatures for two cases: (a) Temperature corresponding to the average of the maximum and minimum impact energies. (b) Temperature at which the impact energy is 20 J. For case (a), we have the average impact energy as 38.75 J. Look for the two data points that have impact energies closest to 38.75 J in the given data. Those would be (30 °C, 71 J) and (20 °C, 58 J). Use linear interpolation to find the corresponding temperature for 38.75 J: Temperature \(\displaystyle = \textnormal{T1} + \dfrac{\textnormal{Impact Energy - E1}}{\textnormal{E2 - E1}}*(\textnormal{T2 - T1})\) \(\displaystyle = 30 \ +\dfrac{38.75 - 71}{58 - 71}*(20 - 30)\) \(\displaystyle\approx 25.44^{\circ} \textnormal{C}\) Therefore, the ductile-to-brittle transition temperature corresponding to the average of the maximum and minimum impact energies is around 25.44 °C. Now, for case (b), Temperature at which the impact energy is 20 J. Look for the two data points that have impact energies closest to 20 J in the given data. Those would be (10 °C, 38 J) and (0 °C, 23 J). Use linear interpolation to find the corresponding temperature for 20 J: Temperature \(\displaystyle = \textnormal{T1} + \dfrac{\textnormal{Impact Energy - E1}}{\textnormal{E2 - E1}}*(\textnormal{T2 - T1})\) \(\displaystyle = 10 \ +\dfrac{20 - 38}{23 - 38}*(0 - 10)\) \(\displaystyle\approx 2.8^{\circ} \textnormal{C}\) Therefore, the ductile-to-brittle transition temperature at which the impact energy is 20 J is around 2.8 °C.

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