4.2 Fatigue
4.2.5 Fatigue Crack Propagation
This master Goodman diagram is useful for predicting fatigue life with any two independent parameters. If the mean stress is 80 ksi and R = 0 (or A = 1), the estimated fatigue life for a smooth unnotched specimen can be predicted at the intersection of the two lines of constant σm = 80 ksi and R = 0 (or A = 1). In this case, the estimated fatigue life will be less than 104 cycles.
The superimposed curves and scales make Figure 4.7 a very busy diagram.
However, the complexity also allows predictions of fatigue life to be made using alternate data sets. The maximum and minimum stresses, σmax and σmin, are the two most commonly cited stress data in fatigue analysis. Their inclusion as the ordinate and abscissa coordinates in Figure 4.7 makes the fatigue life analysis easier, particularly for those who are not familiar with the terminologies used in fatigue analysis, such as R or A ratio. If a notched specimen is subject to a minimum stress, σmin = 20 ksi in a fatigue loading condition with R = 0.2, the corresponding maximum, mean, and alternating stresses will be σmax = 100 ksi, σm = 60 ksi, and σa = 40 ksi, respectively. The fatigue life can be estimated as approximately 3 × 104 cycles by locating the intersection of the two lines of constant σmin = 20 ksi and R = 0.2 in Figure 4.7.
Alternatively, the same fatigue life can also be found by locating the inter- section of the two lines σmin = 20 ksi and σmax = 100 ksi, or σm = 60 ksi and σa = 40 ksi, etc., because all these lines intersect at the same point.
striations can only be observed under a high-magnification microscope.
Fatigue striations should not be confused with benchmarks, which appear as irregular ellipses or semiellipses and can be observed without a microscope.
Benchmarks are also referred to as clamshell marks or arrest marks, as sche- matically illustrated in Figure 4.8. These marks are created by drastic stress changes during fatigue that cause severe deformation and alter crack growth rate. Benchmarks usually converge to the origin of the fracture, which helps determine the location of the crack initiation site. Figure 4.9a and b (NTSB, 2005) shows the fractography of a mechanical part failed due to fatigue in the adjacent area of two drilled holes, as marked by the arrows. The benchmarks emerging from the crack origin are clearly visible on the fracture surface in Figure 4.9a. Figure 4.9b is a close-up view of the fatigue initiation site. The two brackets indicate fatigue origin areas at the surfaces of the fastener hole, and the dashed lines indicate the extent of the fatigue region.
After initiation, a fatigue crack propagates slowly in the order of ang- stroms in the early stage, and shows featureless fracture surface. The prop- agation rate increases to a few microns per cycle after reaching the steady state. For ductile metals such as beta-annealed Ti–6% Al–4% V alloy, the fracture surface generated in this stage typically shows distinctive fatigue striations. However, the presence of striations is not the defining condition for fatigue crack propagation. Many brittle alloys fail by fatigue show- ing no striations at all, and others show striations only in certain areas, as illustrated in Figure 4.10. It is the fatigue fractography of an alumi- num alloy failed after 2.8 × 106 cycles. The crack propagation rate rapidly increases in the final stage, quickly becoming unstable and resulting in a final total fracture.
Figure 4.11 is a schematic representation of fatigue crack propagation rate.
The fatigue crack propagation or growth rate, da/dn, is most often plotted against the range of stress intensity factors ΔK. The stress intensity factor, K, is a measurement of fracture toughness. The maximum, the minimum, and the range of stress intensity factors involved in fatigue crack growth
Crack initiation
site
Bench/clamshell marks
FIgurE 4.8
Schematic of fatigue benchmarks.
are defined by Equations 4.14 to 4.16, respectively, for thin plates with edge cracks under tension.
Kmax= σmax πa (4.14)
Kmin= σmin πa (4.15)
∆K K= max−Kmin (4.16)
(a)
(b)
FIgurE 4.9
(a) Fractography of a fatigue failure. (b) Close-up view of fatigue initiation site. (Both photos courtesy of NTSB.)
Figure 4.11 shows three distinctive regions of fatigue crack propagation behavior. In region I, the fatigue crack does not propagate when ΔK is below a critical threshold value, ΔKth. In region II, a linear empirical relationship as expressed by Paris’ law in Equation 4.17 exists between da/dn and ΔK in the logarithm scale.
da
dn=C( K)∆ p (4.17)
where n is the number of cycles, and C and p are empirical constants. The value of p is approximately 3 for steels, and 3 to 4 for aluminum alloys. Paris’
FIgurE 4.10
Fatigue fractography of an aluminum alloy.
=
Region III accelerated crack growth, unstable
Range of stress intensity factor, ∆K dn
da
ΔKth Region I crack initiation
Region II steady crack growth rate
C(∆K)p dn
da
FIgurE 4.11
Schematic of fatigue crack growth rate.
law offers an important linkage between fatigue phenomena and fracture mechanics through fatigue crack propagation rate, da/dn, and the range of stress intensity factor, ΔK. Region III is a highly unstable region where the crack propagates at an accelerated rate.
The effects of grain size on fatigue life depend on the deformation mode.
Grain size has its greatest effect on fatigue life in the low-stress, high-cycle regime, in which slip band crack propagation predominates. In high stack- ing-fault energy materials, such as aluminum, cell structures develop readily and they control the slip band cracking propagation. As a result, the disloca- tion cell structure masks the influence of grain size, and the grain size has less effect on fatigue life. However, in the absence of cell structure because of planar slip in low stacking-fault energy materials, such as α brass, grain boundaries will control the rate of fatigue cracking. In this case, the fatigue life, Nf, is inversely proportional to the square root of the grain size, as shown in Equation 4.18.
Nf ∝ 1
grain size (4.18)
In general, the fatigue strength of metals decreases with increasing tem- perature with only a few exceptions, for example, mild steel. Fine grain size often results in better fatigue properties at low temperatures. As the temperature increases, the difference in fatigue properties between coarse and fine grain materials decreases. When the temperature reaches a value that is about half the melting point, creep becomes the predominant mecha- nism in determining material strength. Coarse grain materials have higher creep resistance and become stronger. At elevated temperatures, the fracture mechanism will also shift from transgranular, which is typical for fatigue failure, to intergranular, which is typical for creep failure.
4.2.6 Thermal Mechanical Fatigue and Fatigue