Three identical fatigue specimens (denoted A, B, and C) are fabricated from a nonferrous alloy. Each is subjected to one of the maximum-minimum stress cycles listed in the following table; the frequency is the same for all three tests. $$ \begin{array}{ccc} \hline \text { Specimen } & \sigma_{\max }(\boldsymbol{M P a}) & \sigma_{\min }(\boldsymbol{M P a}) \\ \hline \mathrm{A} & +450 & -350 \\ \mathrm{~B} & +400 & -300 \\ \text { C } & +340 & -340 \\ \hline \end{array} $$ (a) Rank the fatigue lifetimes of these three specimens from the longest to the shortest. (b) Now justify this ranking using a schematic \(S-N\) plot.

The stress amplitude \((\sigma_a)\) and mean stress \((\sigma_m)\) for each specimen can be calculated using the following formulas: $$ \sigma_a = \frac{(\sigma_{max} - \sigma_{min})}{2} $$ and $$ \sigma_m = \frac{(\sigma_{max} + \sigma_{min})}{2} $$ Calculate the stress amplitude and mean stress for each specimen: Specimen A: $$ \sigma_{aA} = \frac{(450 - (-350))}{2} = \frac{800}{2} = 400~MPa $$ $$ \sigma_{mA} = \frac{(450 + (-350))}{2} = \frac{100}{2} = 50~MPa $$ Specimen B: $$ \sigma_{aB} = \frac{(400 - (-300))}{2} = \frac{700}{2} =350~MPa $$ $$ \sigma_{mB} = \frac{(400 + (-300))}{2} = \frac{100}{2} = 50~MPa $$ Specimen C: $$ \sigma_{aC} = \frac{(340 - (-340))}{2} = \frac{680}{2} = 340~MPa $$ $$ \sigma_{mC} = \frac{(340 + (-340))}{2} = \frac{0}{2} = 0~MPa $$

To rank the fatigue lifetimes of the specimens, we will compare their stress amplitudes and mean stresses. A lower stress amplitude generally results in a longer fatigue life. Also, having a lower mean stress generally results in a longer fatigue life. From our calculations in Step 1, we can rank the specimens' fatigue lifetimes as follows: 1. Specimen C (Longest fatigue life): This specimen has the lowest stress amplitude (340 MPa) and the lowest mean stress (0 MPa). 2. Specimen B: This specimen has a stress amplitude of 350 MPa and a mean stress of 50 MPa. 3. Specimen A (Shortest fatigue life): This specimen has the highest stress amplitude (400 MPa) and a mean stress of 50 MPa.

An S-N plot is a graphical representation of the relationship between stress amplitude and the number of cycles to failure (fatigue life) for a given material. In an S-N plot, the stress amplitude is plotted on the vertical axis and the number of cycles to failure (in logarithmic scale) is plotted on the horizontal axis. The plot will usually show a downward curve, meaning that higher stress amplitudes result in a shorter fatigue life. To justify our ranking, we can plot the S-N curves for each specimen: 1. Specimen C's S-N curve will be above Specimens B and A, indicating the longest fatigue life. 2. Specimen B's S-N curve will be between Specimens C and A, indicating a medium fatigue life. 3. Specimen A's S-N curve will be below Specimens B and C, indicating the shortest fatigue life. The ranking determined in Step 2 matches the S-N plot description, which confirms our ranking of the fatigue lifetimes of the three specimens from longest to shortest as Specimen C, Specimen B, and Specimen A.