The Fracture Toughness laboratory examined the phenomenon of materials separating or fragmenting into two (or more) pieces under slow loading rate condition. We will now build upon our knowledge of failure mechanisms and study material failures caused by cyclic loading or fatigue. We will use our prior failure knowledge to explain and correlate observed facts from fatigue tests.
One method for studying fatigue problems stems from the area of damage mechanics, commonly known as fracture mechanics. This laboratory builds directly on the Fracture Toughness laboratory you completed earlier. Therefore, it is essential that you review your Fracture Toughness laboratory report, ASTM E399, and ASTM E647 since they are an integral part of this laboratory.
In this laboratory, you are again an engineer in charge of evaluating machine parts design. This time you are assigned to evaluate a part that has been subjected to cyclic loading. You are to determine how many days this part will last until it fails catastrophically. The part has been modeled as a column with through the thickness edge crack as shown in Figure 5.1. You will be performing a fatigue crack growth experiment according to ASTM E647 to aid your investigation. The following information is known:
- The width, W, is 0.1100 meters and the thickness, b, is 0.0254 meters.
- The crack length, a_{o}, is currently 0.0007 m in length.
- σ_{max} is 200 Mpa (assumed upon loading). σ_{min} is 20 Mpa (static weight between loads). Therefore R = 0.1.
- 125 cycles occur every day.
Figure 5.1
Figure 5.1 Industrial machine part with a through the thickness edge crack.
- The stress intensity factor, K, can be determined from K = f(α) σ sqrt[π a]
- For the case of edge-cracked plate loaded in tension,
We will assume that a « W and neglect all a terms.
The tasks listed below outline the necessary steps to carry out the fatigue crack growth test. It also points out factors effecting your experimental result that must be addressed in your report. See the Appendix for help with the use of your experimental result in your analysis.
Tasks
- Test a compact tension (CT) specimen according to ASTM E647. Calculate (if necessary) and include the raw a vs. N data in your report (a is the crack length and N is number of cycles). In addition, include the Fatigue Precrack Data Report Form if available.
- Create a plot showing a vs. N. Create another plot showing a vs. time.
- Calculate da/dN and ΔK values from the a and N data collected above. Calculate da/dN using the forward difference approximation:
This value of da/dN is equal to the average crack growth between stages i and i+1. Since da/dN is an average computed over the a_{i} +1 - a_{i} increment, use the average crack length over the region when calculating ΔK. The average crack length, a_{avg}, can be determined from:
For the CT specimen:
Where,
a = a_{avg}/W
ΔP = load range
B = specimen thickness
W = specimen length
- Plot log da/dN vs. log ΔK. Note any trends.
- The Paris law (shown below) is a popular method and is frequently used to represent a portion of the fatigue-crack growth curve. Would this equation generally be applicable over the entire crack growth curve? (To help answer this, look at your da/dN vs. ΔK plot and work the rest of the Tasks.)
- Perform a linear least squares cure fit on your data to determine the Parameters C and n for Paris law. Remember to only curve fit the linear portion of the data shown in the plot from Task 4
- On the second plot from Task 2 (a vs. time), mark the starting and ending points of the data used to calculate the C and n on the time axis. Comment of what region the Paris law covers on a time basis (e.g., 90% of the time range, 40% of the time range, etc.).
- If you look at your plot from Task 4 you will notice three Regions similar to those shown in Figure 5.2. In the first region, the slope is steeper than that of the linear second region. Therefore, if you use the Paris relationship for crack lengths in this region with the C and n values determined from the second region, you will under predict the crack growth rate. The same is also true for Region III.
Figure 5.2
Figure 5.2 Three regions of crack growth rate curve used for the discussion.
- Comment on the usefulness of the Paris law for predicting fatigue crack growth data and structural failure.
- Many engineers look-up and use the Paris law from handbooks without knowing much about it. Do you think this is a good idea? Is it better than nothing? Does the Paris law work well if you know the limitations? Explain your answer in detail.
- Comment on the similarities and differences in predicting failure for quasi-static loading type of failures (plastic strain, instability, buckling, fracture toughness, etc.) vs. dynamic loading type of failures (fatigue).
- Calculate the starting and stopping ΔK values for your machine part. Determine whether you are in the region where Paris relationship is valid. What region are we in initially and at failure? What does that tell you about your analysis? How much time is spent in Region III anyway?
Appendix
Using experimental fatigue crack growth data to predict failure.
You need to determine the critical size of the crack,
a_{crit}, that will cause failure. First uses the Δ
K_{max} value you determined from the experiment to determine
K_{Ic}. We are treating the
K_{Ic} as a material property but hopefully you learnt the limitation to this assumption from previous lab.
Then using
K_{Ic}, solve for
a_{crit} for your machine part. The equation for the stress intensity factor still holds, therefore, you can calculate
a_{crit}for your machine part. Remember to use the appropriate correction factor.
With the critical crack size known, we have to integrate the Paris law to determine the number of cycles,
N_{f}, until failure.
C and
n values are the values you determined from the laboratory but use the expression for Δ
K that is appropriate for your machine part.
(Think about how much more work would be involved in integrating a more complicated Δ
K equation such as that of the CT specimen.)