The following tabulated data were gathered from a series of Charpy impact tests on a tempered 4340 steel alloy. $$ \begin{array}{|cc|} \hline \text { Temperature }\left({ }^{\circ} \boldsymbol{C}\right) & \text { Impact Energy (J) } \\ \hline 0 & 105 \\ \hline-25 & 104 \\ \hline-50 & 103 \\ \hline-75 & 97 \\ \hline-100 & 63 \\ \hline-113 & 40 \\ \hline-125 & 34 \\ \hline-150 & 28 \\ \hline-175 & 25 \\ \hline-200 & 24 \\ \hline \end{array} $$ (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 \(50 \mathrm{~J}\).

Step by step solution

01

(Plotting the Data)

For part (a), you need to plot the data as impact energy versus temperature. Using software like Excel, MATLAB, or a graphing calculator, input the given data points and create a scatter plot with temperature on the x-axis and impact energy on the y-axis.

02

(Finding the Average of Maximum and Minimum Impact Energies)

For part (b), first, we need to find the maximum and minimum impact energies from the given data. The maximum impact energy is 105 J, and the minimum impact energy is 24 J. Now, we will calculate the average of the maximum and minimum impact energies: $$ \text{Average} = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{105 + 24}{2} = 64.5 \text{ J} $$ Now, we need to find the temperature that corresponds to this average impact energy. We observe that the impact energy of 64.5 J falls between the data points (-75, 97) and (-100, 63). We'll interpolate between these points in the next step.

03

(Interpolating for Temperature)

To interpolate the temperature corresponding to an impact energy of 64.5 J between the points (-75, 97) and (-100, 63), we'll use the equation for interpolation: $$ \text{Temperature} = \frac{\text{Impact Energy} - \text{Impact Energy}_1}{\text{Impact Energy}_2 - \text{Impact Energy}_1} \times (\text{Temperature}_2 - \text{Temperature}_1) + \text{Temperature}_1 $$ For our case: $$ \text{Temperature} = \frac{64.5 - 97}{63 - 97} \times (-100 + 75) + (-75) = -84.8 ^{\circ} \text{C} $$ Hence, the ductile-to-brittle transition temperature corresponding to the average of the maximum and minimum impact energies is approximately -84.8°C.

04

(Finding the Temperature with Impact Energy of 50 J)

For part (c), we need to determine the ductile-to-brittle transition temperature as the temperature at which the impact energy is 50 J. Since this value lies between the data points (-75, 97) and (-100, 63), we'll use the interpolation formula again: $$ \text{Temperature} = \frac{50 - 97}{63 - 97} \times (-100 + 75) + (-75) = -92.1 ^{\circ} \text{C} $$ Hence, the ductile-to-brittle transition temperature at which the impact energy is 50 J is approximately -92.1°C.

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