MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II HOMEWORK # 6 Distributed: Wednesday, April 14, 2004 Due: Wednesday, April 21, 2004 Problem 1 A fatigue crack growth test was done on a compact tension specimen of a hard (HRC=60) tool steel. A table of respective values of stress intensity factor range, ΔKI , and correspond-ing fatigue crack growth rates, da/dN , are given in the table. da/dN (mm/cycle) 4.26 × 10−6 9.12 × 10−6 1.75 × 10−5 3.51 × 10−5 ΔKI (M P a√m) 6.84 8.76 10.35 13.3 • Plot these points on log-log coordinates (over a suitable range!) and graphically esti-mate values of the “Paris-law” fitting constants A and m that describ e fatigue crack growth in this material according to da = A (ΔKI )m . dN Be sure to give appropriate units! • Use a least squares fit to the dataset to obtain refined values for A and m. • For the least squares best fit to the Paris law constants, describ e how you would obtain an alternative set of parameters Δa0, ΔKI0, and m that equivalently describe fatigue crack growth according to � ΔKI �mda = Δa0 ,dN ΔKI0 and give numerical values for all parameters. (Based on Dowling text, problem 11.3). 1Problem 2 An edge crack of initial length ai = 3mm exists in a large plate that is to be subjected to a remote cyclic stress, σ∞, ranging from σmin = 0 to some to-be-determined maximum value of σmax (R = 0). You may assume that the plate is sufficiently large that the configuration correction factor Q = 1.12 in the expression for KI for all crack length values, a, to be considered. The fracture toughness of the material is KIc = 115 MP a√m and its yield tensile strength is σy = 1045 MP a. Fatigue crack growth is describ ed by a Paris law form � ΔKI �mda = Δa0 ,dN ΔKI0 with m = 4, Δa0 = 10−5 mm/cycle, and ΔKI0 = 20 MP a√m. It is desired to be able to apply N = 50, 000 load cycles to the pre-cracked structure while still retaining a factor of safety of at least 2 on the actual fatigue crack propagation life until fracture. That is, it is desired that the predicted fatigue crack propagation life for the chosen value of σmax be at least ≥ 100, 000 cycles. • What is the largest value of σmax that meets this constraint? Note: this problem is a bit “non-standard”, in that the upper limit of the fatigue crack length, af , depends explicitly on the unknown value of σmax (why is this so?), but so does the number of cycles, Nai af , required to propagate the crack. You may find →it desirable to manipulate the relevant equations(s) in a form suitable for numerical iteration or graphical solution. • For your chosen value of σmax , make a graph of the crack length, a(N), for 0 ≤ N ≤100, 000 cycles. What is the predicted value of crack length at N = 50, 000 cycles, a(N = 50, 000), and what is the corresponding value of KI? Based on this KI-value, what is the predicted factor of safety with respect to fracture at this point (i.e., what is the ratio of KIc/KI(σmax , a(N = 50, 000)))? • What conclusions do you draw about the distinction between factors of safety on fatigue crack propagation life and on fracture itself ? Discuss.
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