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2 Maximum stress = σ max (Eq. 1) Minimum stress = σ min (Eq. 2) Mean stress = σ mean = ( σ max + σ min ) / 2 (Eq. 3) Stress Amplitude= σ amp = ( σ max - σ min ) / 2 (Eq. 4) Stress Ratio R = σ min / σ max (Eq. 5) Figure 1: Fully Reversed Stress (σ) of a single point on a specimen 2.3 S-N Curves The goal of this experiment was to find the endurance limit of round axle like specimens made of SAE 1045 cold rolled steel. By applying a bending moment to the specimen and then rotating the specimen, points on the surface go from compression to tension and back to compression. This happens once for every revolution and is similar to bending the bar back and forth. The difference being that the stress is applied to every point on the surface. By varying the maximum stress level that is applied to the bar and counting the number of revolution until failure, data points are created. These points are then plotted forming a curve which is termed an S-N curve. This curve is used to predict when if ever a part will fail. An example curve is seen in Figure 2 where the x axis represents the number of cycles and the y axis represents the magnitude of maximum and minimum applied stress. Notice in Figure 2 the stress at which Figure 2: Example S-N Curve

3 the curve flattens out. This stress is termed the endurance limit (S e ). Stress below this point will not cause failure. 2.4 Stress in a round beam Figure 3: ANSYS simulation of beam in pure bending This experiment was conducted using a round shank specimen. Figure 3 above is a simulation of the strain that the specimen undergoes. Notice the maximum strain values at the top and bottom of the reduced section. Also notice that the strain and thus the stress are close to 0 at the center. This implies that as the specimen rotates the stress at each point changes from tension to compression and back to tension again. This happens once for each rotation. The 0 stress/strain line in the case of the beam is termed the neutral line. Stress can be calculated by using Equation 6 where sigma is the stress M is the moment y is the distance from the center line to the outside surface and I x is the area moment of inertia. The magnitude of the stress is the same at both the top and bottom and can be calculated using the Equation 6. (Eq. 6) The moment M in Equation 6 is calculated using Equation 7 where W is the weight applied and L is the length of the specimen. (Eq. 7) The area moment of inertia I x in Equation 6 is calculated using Equation 8 where r is the radius of the specimen. (Eq. 8) Equations 6 through 8 can be combined and simplified to form Equation 9 where W is the applied weight, L is the moment arm, which for this experiment is always 4, and D is the smallest diameter of the specimen.

4 (Eq. 9) Equation 9 gives the maximum stress at the surface. This stress is the stress that is plotted on the y axis of the S-N curve (Figure 2). 2.5 Crack Propagation It is assumed that as long as a material is not stressed beyond its yield point it will not fail. Note, however, that materials are not perfect and these imperfections affect the strength of a material. Small cracks can develop in the surface of a material, called stage I fatigue, where the stress is the highest. These cracks can grow with the repeated application of stress to which is referred to as stage II fatigue. Each time the stress is applied the crack grows a little more. At some point this crack reaches a critical point and causes a catastrophic failure in the material termed stage III fatigue. This Crack propagation, however, does not happen at just any applied stress but instead at a critical stress. This is because cracks have points on the end which act as stress risers. If the stress is not high enough then the stress will not rise high enough for the ends to separate. The speed of propagation is also dependent on how high the applied stress is. Thus, the number of cycles that are needed to cause failure varies if failure happens at all. In the S-N diagram (Figure 2) first notice the line for steel where each point on the line represents failure. Notice that as stress goes up, the number of cycles is reduced. However, what happens at a stress below 30 ksi? At and below this stress, failure does not occur. This point is called the endurance limit. As alluded to previously, applied stress below this point does not cause crack propagation and therefore does not cause failure. Notice also that in the more ductile aluminum there is no perceptible endurance limit. This implies that sooner or later aluminum will fail under a cyclical load. This experiment however was concerned with steel. It should also be noted here that the fracture that occurs is not ductile fracture. Crack propagation whether granular or trans-granular leaves a more horizontal break with very little area reduction. Thus, observing a break that has undergone brittle fracture should appear similar to Figure 4 below. Notice in the figure that there is very little noticeable plastic deformation or reduction in the cross sectional area. A closer look at a brittle fracture reveals the faceted surface characteristic of brittle fracture. The difference between ductile and brittle fracture can be seen in Figure 5. Figure 4: Brittle fracture of Aluminum Figure 5: (a) Brittle fracture (b)(c) Ductile fracture

5 2.6 Pure Bending Here it is important to talk about pure bending. To reiterate, Equation 9 does not take into account any type of torsion, or tensile forces that may be on the specimen. It only considers stress due to moments placed on the ends of the specimen. So how is this type of bending achieved? Truly it is impossible. Imagine if a force perpendicular to the axis was placed on a fixed bar to cause bending as seen in Figure 6. The applied force would pull or push down on the bar and create axial stress on the bar as well as a bending moment. However, the R.R. Moore machine is designed to approximate pure bending with no axial strain. Figure 6: Fixed beam under bending and axial stress It does this by placing weights on each end of the bar but allows the mounts to float. The mounts float but they also rotate. Notice in Figure 7 that the arrows represent where the weight is applied, one for each end, while the dots are rotation points. So as the weight pulls down, the bearing housings into which the sample is loaded are allowed to slide horizontally and rotate at the same time. This relieves axial tension and leaves only a pure bending moment on the sample. Figure 7: R.R. Moore Rotational Beam Fatigue Testing Machine 2.7 Speed of rotation and Heat Treatment of Samples Several questions may arise when performing this type of experiment. First is, how does the speed of rotation affect the fatigue strength? Second is, how does heat treating the specimen affect the fatigue strength? It is logical to theorize that the faster the speed the lower the tolerance for stress would be. This would be due to the loss of ductility due to the higher speed as dislocations would have less time to travel past obstacles. With a heat treated sample there are three possibilities that should be noted. First is that the decreased ductility of the steel would further encourage crack propagation. Second is that the increased hardness would strengthen the surface and therefore stress levels for crack propagation would be higher. Third is that heat treatment would have no effect; which is unlikely.

6 3. Sample Preparation This experiment requires the sample to be absolutely centered in the machine. For this reason the collets which are supplied with the machine require the specimen to be used, to be within certain tight tolerances. For the collets that were used in this experiment the diameter of the ends of the specimen was specified as inches. The specified tolerance is / inches. It is important to follow these guidelines so as not to damage the precision collets. The diameter at the center length of the samples can vary. For this experiment the diameters of the samples ranged between to inches. It is also important to note that the finish on the sample should be between a 16 to 32 finish. This is important because surface flaws, especially in the radial direction act as cracks and can cause premature failure. Appendix A contains sample drawings. The product manual also contains specimen drawings. It should be noted in this experimental study that heat treated specimens were prepared by placing them in an 850 o C oven for 30 minutes. They were then removed and allowed to air cool.

7 4. Procedure This procedure assumes that there is no specimen in the machine. 1. First make sure the machine is not plugged in. Premature starting of the machine could damage the machine or even cause injury. 2. Open the Plexiglas enclosure by rotating it about the hinges at the middle of the machine. 3. Remove each of the bearing housings by lifting them straight up and out of the machine and set them down on the bench. The roller supports fit tightly between the rails and thus will jam against the sides very easily. Take care not to twist them in anyway while removing. 4. Now take the specimen that will be used and slide a collet onto each end. The collets should slide on tightly but without much effort. If they are two difficult to insert they may be out of tolerance. 5. Next slide one of the collets into the left bearing housing (assuming the motor assembly is on the right). On the other side of the bearing housing insert the draw-bar (the one without the set screw in the head) and slowly tighten until finger tight. Notice that collet is being drawn into the housing. Now using the supplied wrenches hold the spindle tight with one and with the other turn the drawbar approximately another half rotation or until tight. 6. Now repeat the process with the right bearing housing. Make sure to use the draw bar with the set screw. Once the draw-bar is tight the assembly is ready to be placed back into the machine. 7. Lift the entire assembly by grasping both bearing housings and gently lower the housing rollers onto the rails. Again note the tight tolerances and keep it very straight. Also take care not to damage the flexible spindle attached to the motor on the right side. 8. Next the motor must be connected to the bearing assembly. Carefully slide the flexible spindle into the provided hole on the end of the draw-bar. Note that the spindle has a flat side onto which the set crew should land when tightened. Align the set screw with the flat side of the spindle and tighten using the supplied Allen-wrench. 9. Now the Plexiglas cover should be closed. Note that if it is opened during the test the machine will stop. This could damage the sample and the test will have to be redone. 10. The machine can now be plugging in. 11. Make sure to reset the counter before the start of each test. 12. Now note that there is a small reflective strip attached to the spindle mount coming out of the motor assembly. This strip in conjunction with the hand held tachometer allows for accurate measurement of the RPM the machine is running at. 13. Also note the speed control knob on the top of the machine. This knob changes the voltage to the motor and thus the speed of the motor can be controlled. This knob in conjunction with the tachometer can be used to set the speed of the machine. 14. Once the machine is plugged in and the counter reset the start button should be pushed 15. Now that the machine is running point the tachometer at the now rotating reflective strip. Push the test button on the tachometer and read the output. The tachometer may have to be moved around a bit to get an accurate reading. It is okay to press the tachometer against the Plexiglas to gain stability. Once a reading is taken the speed can be adjusted to reach the required RPM. The RPM should be checked throughout the test and the speed adjusted accordingly. 16. Once the specimen breaks the total revolutions should be noted. 17. In order to remove the specimens reverse the above procedure. Note that once the draw bars are removed use the extractor bolt to force the collets out of the housings. Do this by screwing the extractor bar into the bearing housing until the collets are released. 18. The specimens can now be easily removed from the collets

8 5. Experimental Analysis 5.1 Graphical Analysis As can be seen in Table 1 below, tests were run at three different speeds, 2000, 3000 and 4000 RPM. There was also a test run on a heat treated set at 3000 RPM denoted as 3000HT in the RPM column. The applied stress (σ), specimen diameter (D), and number of rotations (N) were noted for each test. Stress (σ), was calculated using Equation 9. Note that several of the samples did not break within a reasonable number of rotations and the tests were stopped and the data was not used. Table 1: Diameter, D, Effective Load, W, Stress, σ, Total Revolutions (N) Test # D, Diameter (in) W, Effective Load (lbs) N, Total Revolution σ, Stress RPM Test # Diameter (in) Effective Load (lbs) Total Revolution (N) Stress (σ) RPM , Test # Diameter (in) Effective Load (lbs) Total Revolution (N) Stress (σ) RPM HT HT HT HT HT Test # Diameter (in) Effective Load (lbs) Total Revolution (N) Stress (σ) RPM The data in Table 1 was then graphed as seen in Figure 8 and Figure 9. In Figure 8 the S-N curves of the 2000, 3000 and 4000 RPM tests were compared. The Y axis is again the stress denoted as S or σ, while the X axis is the number of rotations, N.

9 Figure 8: Comparison of specimens based on Speed of Rotation It can be seen in Figure 8 that the speed of rotation did affect the shape of the curve. The shape of the curve directly correlates to the overall fatigue strength. That is to say, the lower the curve the lower the overall fatigue strength of the material at that speed. The graph, therefore, shows that fatigue strength is inversely related to the speed at which the sample is run. Table 2 lists the trend line equations for trend lines seen in Figure 8. These equations can be used for predicting failure at the stated RPM. The validity of the equations is verified by the listed R 2 values. Table 2: R 2 values and empirical equations for graphs in Figure 8 Rotation Speed (RPM) Line fit equation R 2 Value 2000 σ = N σ = N σ = N In the second graph, Figure 9, the specimens tested at 3000 RPM were compared with heat treated specimens. It can be seen that the heat treatment had the effect of lowering the fatigue strength of the material. To verify this result note the dashed horizontal lines drawn at similar stress levels. It can be seen that for samples run near these stress levels the number of cycles to failure of the heat treated samples is lower.

10 Figure 9: Comparison of cold rolled and heat treatment specimens Table 3 lists the trend line equations for trend lines seen in Figure 9. These equations can be used for predicting failure at the stated RPM. The validity of the equations is verified by the listed R 2 values. Table 3: Empirical equations and R 2 values at 3000 RPM, for cold-rolled and Heat Treated data sets as graphed in Figure 9 Rotation Speed (RPM) Line fit equation R 2 Value 3000 σ = N Heat treated σ = N

11 Another important result that should be noted is the type of fracture that occurred. It can be seen by looking at Figures 10 and 11 that the fracture was brittle in all cases as there is no sign of ductile fracture as is seen in regular axial tensile testing. Blurriness of the image (Figure 11) in parts can be attributed to two things. First is the asperity of the surface. Second is the inability of the microscope to focus on all parts of the surface at the same time due to the aforementioned surface asperity. Thus parts of the surface that are out of focus are either much lower or much higher than the focused areas. Figure 10: Brittle Fracture Surface Figure 11: Brittle Fracture surface (20X magnification) 6: Conclusion Two theories were put forth at the beginning of this experiment. First was that at higher cyclic speeds the specimens would exhibit a lower fatigue strength. It can be seen that the curves in Figure 8 do support the initial hypothesis with the 2000 RPM curve being higher than 3000 RPM; both of which are higher than the 4000 RPM curve. This is due to a reduction in material ductility as higher cycle speeds are used. This therefore increases stress at the risers (corners of the crack) and encourages crack propagation. Heat treatment was shown to reduce cyclic fatigue strength. This is due to the overall decrease in ductility of the material. Where it was proposed that the hardening of the surface may increase the fatigue strength it turns out that the overall increase of the brittleness of the steel was the major factor. That is to say the material became too brittle for the hardened surface to make much of a difference. It should be noted that in Figure 2 the very ductile aluminum does not show an endurance limit. This is due to its ductility. Thus if, in steel, this ductility could be maintained and the surface hardened then a different outcome may have been attained. This could have been accomplished through carbonizing of the surface. Finally, this experiment has yielded two significant results. First is that increased speed of applied cyclic stress decreases the fatigue strength and increases the likely hood of catastrophic brittle fracture. The second conclusion that can be made is that heat treatment of 1045 steel increases the brittleness of the material so as to overshadow any surface hardness effects that may have been beneficial. It is therefore very important when designing system components to take into account both the life (number of cycles to failure) and the range of the speed of the applied stress that the part will undergo. It is then also important to know that heat treatment will not necessarily improve endurance and in fact may have a significantly detrimental effect on the part.

12 7. Acknowledgements Author Daniel Hendrickson would like to thank the University of Michigan-Flint Department of Mechanical Engineering, for the use of the R.R. Moore machine as well the laboratory facilities. The author would also like to acknowledge Dr. O. Aluko for his guidance and support during this experiment and John O Brien for his guidance in sample preparation and machine setup. 8. References Budynas, R., Nisbett, J. & Shigley, J. (2011). Shigley's mechanical engineering design. New York: McGraw-Hill. Vable, M. (2002). Mechanics of materials. New York: Oxford University Press.

13 9. Appendix 9.1 Schematic Drawing of Specimens (see attached)

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