MECH4900 Final Assignment T22024
The purpose of this assignment is to apply the knowledge learned in MECH4900 to understand problems that a practicing engineer might face in their career. This will include:
1) Reading two assigned journal articles,
2) Completing a very short multiple choice (no calculations) Moodle quiz, and
3) Submitting an assignment of questions and calculations.
The final assignment reading quiz will be open Thursday 1 August until Friday 2 August at 23:59. The quiz questions are meant to assess if you read the articles and case studies carefully enough to understand the concepts. The quiz is meant to be a relatively easy part of the final assignment.
The assignment is to be typed and uploaded to Moodle by Friday 2 August at 23:59, links to Moodle are also available on MS Teams. When using mathematics software, please insert the relevant sections (screenshots, PDFs, etc.) in each problem or attach your code/notebook (with references to problem numbers) to your assignment.
Problem set 1: Fatigue lifetime analysis based on different sample orientations.
Download from Teams/Moodle and read the article:
· Borges MF, Antunes FV, Jesus J, Branco R, Prates P, Neto DM. “Effect of LT and ST orientations on fatigue crack growth in an aluminum 7050T7451 plate.” Fatigue Fract Eng Mater Struct. 2023; 46(9): 32743289. https://doi.org/10.1111/ffe.14067
We learned for K_{IC} testing how the sample orientation can influence the fracture toughness, and in this journal article we learn how it may also affect the fatigue crack growth rates and the predicted fatigue lives. This article also discussed other factors that may affect fatigue crack growth such as crack closure and smallscale yielding conditions. In this problem set, we will examine some of these issues based on the above journal article.
Problem set 2: Nonstandard testing methods for fracture toughness
Download from Teams/Moodle and read the article:
· Moschetti, M. et al. Fracture Toughness Investigations of an IonIrradiated Nanocrystalline TiZrNbHfTa Refractory HighEntropy Alloy. Adv. Eng. Mater. 2024; Article 2400541. https://doi.org/10.1002/adem.202400541
It is often necessary to use or develop nonstandard test methods when standards simply do not exist for your situation. In some cases, practicing engineers and scientists use nonstandard test methods when the amount of available material is much smaller than any standard would permit. However, despite our best attempts, the results can be highly inaccurate and misleading. In this journal article, we attempted to use the popular microcantilever fracture toughness test to study radiation embrittlement of a new type of alloy. While the effect of radiation embrittlement was clear from our study, accurate fracture toughness values were not achieved by that test. In this problem set, we will examine some of the relevant issues.
Problem Set 1: Fatigue lifetime analysis based on different sample orientations.
Borges MF, Antunes FV, Jesus J, Branco R, Prates P, Neto DM. “Effect of LT and ST orientations on fatigue crack growth in an aluminum 7050T7451 plate.” Fatigue Fract Eng Mater Struct. 2023; 46(9): 32743289. https://doi.org/10.1111/ffe.14067
Based on what we know about fracture toughness, we should expect K_{IC} to be different for the LT and ST orientations. The K_{IC} values for 7050T7451 aluminum are given as:
Properties for
thick 7050T7451 aluminum

Value (with units)

Average K_{IC} LT Orientation

35.2 MPa√m

Average K_{IC} ST Orientation

28.6 MPa√m

Let’s examine how the crack orientation will affect the fatigue lifetime for aluminum alloy 7050T7451. In Equation 1, the authors show the stress intensity factor for the laboratory specimens they used to measure fatigue crack growth rates. In real applications, aluminum alloy 7050T7451 is used in large aerospace structures such as fuselage frames and wing skins making it more practical to use infinite plate K_{I} solutions.
1) Assuming a through thickness crack in an infinite plate, calculate the critical crack size, a_{c}, for both the LT and ST orientations assuming an initial crack size of a = 1 mm, a cyclic stress range of ∆�� = 114 MPa, and a stress ratio of R = 0.05.
2) Assuming a through thickness crack in an infinite plate, used the experimental results in Figure 2 of Borges et al. to calculate the fatigue life for both the LT and ST orientations assuming an initial crack size of a = 1 mm, a cyclic stress range of ∆�� = 114 MPa, and a stress ratio of R = 0.05.
Hint for 2): The Paris law coefficients “C” given in Figure 2 use two different length units, mm on the vertical axis and m on the horizontal axis. To calculate lifetime, we need a single unit for length. I recommend dividing “C” by 1000 to convert the growth rates from mm/cycle to m/cycle.
(Note, for Quizzes and Exams, you will always be given the correct units to use so you don’t need to worry about this issue. But you may encounter this issue in real world analysis!)
Another important consideration influencing the fatigue lifetime is the stress ratio which affects the amount of crack closure. Borges et al. created a numerical model that can predict fatigue crack growth rates different stress ratios. Using their results in Figure 12, recalculate questions 12 assuming a stress ratio of R = 0.5. Assume crack closure does not occur at R = 0.5.
3) Recalculate the critical crack size, a_{c}, for both the LT and ST orientations assuming an initial crack size of a = 1 mm, a cyclic stress range of ∆�� = 114 MPa, but now with a stress ratio of R = 0.5.
4) Recalculate the fatigue life for both the LT and ST orientations assuming an initial crack size of a = 1 mm, a cyclic stress range of ∆�� = 114 MPa, but now with a stress ratio of R = 0.5.
Hint for 4): This time you should divide “C” by 10^{6} to convert the growth rates from μm/cycle to m/cycle. See 2) above for more information.
Finally, the authors Borges et al. point out the importance of the smallscale yielding (SSY) assumption in fatigue crack growth analysis using LEFM. They also point out that, unfortunately, ASTM Standard E647 does not clearly define the bounds for SSY. Thus, Borges et al. look to the literature and evaluate Equation 8 to determine when SSY conditions occur. Based on the results in Fig. 14:
5) Were SSY conditions met at the onset of crack growth for Problem 2 for the LT or ST orientations? Support your answers with a calculation and an explanation to receive credit.
6) Were SSY conditions met at when final fracture occurred for Problem 2 for the LT or ST orientations? Support your answer with a calculation and an explanation to receive credit.
7) Were SSY conditions met at the onset of crack growth for Problem 4 for the LT or ST orientations? Support your answer with a calculation and an explanation to receive credit.
8) Were SSY conditions met at when final fracture occurred for Problem 4 for the LT or ST orientations? Support your answer with a calculation and an explanation to receive credit.
9) Based on your answers regarding SSY, describe some limitations in calculating fatigue lives to failure using LEFM analysis for aluminum alloy 7050T7451. Also explain how you might avoid such limitations in a damage tolerant analysis like we learned in class and like is discussed in the textbook chapter 10.10 in the 4^{th} edition (or chapter 10.9 in 3^{rd} edition).
Problem Set 2: Nonstandard testing methods for fracture toughness
Moschetti, M. et al. Fracture Toughness Investigations of an IonIrradiated Nanocrystalline TiZrNbHfTa Refractory HighEntropy Alloy. Adv. Eng. Mater. 2024; Article 2400541. https://doi.org/10.1002/adem.202400541
10) In your own words, describe why the authors needed to test microscale specimens and why we decided to refine the grain size to nanometer dimensions (i.e., nanocrystalline alloy).
11) In this study, the ASTM standard J_{IC} testing requirements were not met because the samples were too small. Thus, the authors didn’t measure valid J_{IC }or K_{JIC} values and instead estimated the initiation toughness as J_{init} and K_{Jinit}. The authors measured very different K_{Jinit} values for two different undersized sample tests. Suppose you wanted to design a reactor component based on the measured values for unirradiated, nanocrystalline TiZrNbHfTa. Calculate the two different fracture stresses for a 5 mm long edge crack in a very large plate under tension loading based on the two different K_{Jinit} values measured for unirradiated, nanocrystalline TiZrNbHfTa.
12) While both values may be somewhat inaccurate, which calculated fracture stress from 11) would you expect to be closer to reality for a large plate of unirradiated, nanocrystalline TiZrNbHfTa. Provide a logical justification of your answer to receive any marks.
13) Repeat your calculation for 11) based on the reported ASTM standard K_{Jinit} value (note, this is a valid K_{JIC}) for fracture toughness for an unirradiated, coarsegrained plate of TiZrNbHfTa.
14) How do the authors explain the much lower fracture toughness of nanocrystalline versus coarse grained TiZrNbHfTa? Provide at least three contributing factors.
15) Irradiation caused a 27% decrease in the measured fracture toughness for the microscale, nanocrystalline specimens. Real reactor components are made from coarse grained alloys. Based on this manuscript, would you be confident designing reactor components by reducing the ASTM valid fracture toughness for coarse grained TiZrNbHfTa by 27%? Explain why or why not to receive any marks.