Design Procedure 7.1: Methods to Maximize Design Life
7.5 Fatigue Strength
7.5.1 Rotating-Beam Experiments
Fatigue is inherently probabilistic; that is, there is a great range of performance within samples prepared from the same materials. In previous problems and case studies, a valuable approach called the worst-case scenario was de- scribed. To apply this approach to fatigue, a designer would select surface finishes, notch sizes, initial flaw size, etc, that minimize the fatigue strength of the candidate specimen.
However, this process would result in fatigue specimens with zero strength, a situation that does nothing to aid designers.
Thus, data on fatigue often reflect the best-case scenario, and do not reflect actual environments. The designer is strongly cautioned that great care must be taken in applying fatigue design theories based on best case scenarios to critical appli- cations.
Because fatigue is a damage accumulation phenomenon, initial flaws have a large effect on performance. No manufac- turing process produces defect-free parts; indeed, it is not un- common to encounter thousands, even millions, of flaws per cubic millimeter. The flaws are distributed in size, shape, lo- cation and orientation, they are often close enough to violate Saint-Venant’s principal (see Section 4.3), so that the associ- ated stress concentrations interfere with and compound each other. Analytical approaches that derive fatigue strengths from first principles are thus very difficult, and most knowl- edge on material fatigue is experiment-based.
Experimental approaches to fatigue use either exemplars or idealized, standard specimens. The former are more re- liable and best for critical applications. The latter are often
7.5 247R 86
Figure 7.3: R.R. Moore machine fatigue test specimen. Di- mensions in millimeters.
used when a direct simulation of the loading environment is cost prohibitive.
To establish the fatigue strength of an exemplar, a se- ries of tests is performed. The test apparatus duplicates as nearly as possible the stress conditions (stress level, time fre- quency, stress pattern, etc.) in practice. The exemplar dupli- cates as nearly as possible any manufacturing and treatment processes. Such experiments give the most direct indication of a component’s survivability in the actual loading environ- ment.
To test idealized, standard specimens, a rotating-beam fatigue testing machine is often used, such as the Moore rotating-beam machine. The specimen is subjected to pure bending, and no transverse shear is imposed. The specimen has specific dimensions (Fig. 7.3) and a highly polished sur- face, with a final polishing in the axial direction to avoid circumferential scratches. If the specimen breaks into two equal pieces, the test is indicative of the material’s fatigue strength. If the pieces are unequal, a material or surface flaw has skewed the results. The test specimen is subjected to completely reversed (σm = 0) stress cycling at a relatively large maximum stress amplitude, usually two-thirds of the static ultimate strength, and the cycles to failure are counted.
Thus, for each specimen at a specific stress level, the test is conducted until failure occurs. The procedure is repeated on other identical specimens, progressively decreasing the max- imum stress amplitude.
7.5.2 Regimes of Fatigue Crack Growth
Figure 7.4a shows the size of a fatigue crack as a function of number of cycles for two stress ratios. Figure 7.4b illustrates the rate of crack growth, and more clearly shows three differ- ent regimes of crack growth:
1. Regime Ais a period of very slow crack growth. Note that the crack growth rate can be even smaller than an atomic spacing of the material per cycle. Regime A should be recognized as a period of non-continuum failure processes. The fracture surfaces are faceted or serrated in this regime, indicating crack growth is pri- marily due to shear deformations within a grain. The growth rate is so small that crack lengths may be neg- ligible over the life of the component if this regime is dominant. Regime A is strongly affected by material mi- crostructure, environmental effects, and stress ratio,R.
2. Regime Bis a period of moderate crack growth rate, of- ten referred to as theParis regime. In this regime, the rate of crack growth is influenced by several factors, in- volving material microstructure, mechanical load vari- ables, and the environment. Thus, it is not surprising that crack propagation rates cannot be determined for a given material or alloy from first principles, and testing is required to quantify the growth rate.
3. Regime Cis a period of high-growth rate, where the max- imum stress intensity factor for the fatigue cycle ap- proaches the fracture toughness of the material. Ma- terial microstructural effects and loadings have a large influence on crack growth, and additional static modes such as cleavage and intergranular separation can occur.
7.5.3 Microstructure of Fatigue Failures
As discussed above, even the most ductile materials can ex- hibit brittle behavior in fatigue, and will fracture with little or no plastic deformation. The reasons for this are not at all ob- vious, but an investigation of fatigue fracture microstructure can help explain this behavior.
A typical fatigue fracture surface is shown in Fig. 7.5, and has the following features:
1. Near the origin of the fatigue crack (Point B in Fig. 7.5), the surface is burnished, or very smooth. In the early stages of fatigue, the crack grows slowly and elastic deformations result in microscopic sliding between the two surfaces, resulting in a rubbing of the surfaces and associated mechanical polishing.
2. Near the final fracture location (Point A in Fig. 7.5),stri- ationsorbeachmarksare clearly visible to the naked eye.
During the last few cycles of a fatigue failure, the crack growth is very rapid, and these striations are indicative of fast growth and growth-arrest processes.
3. Microscopic striations can exist between these two ex- tremes as shown in Fig. 7.5, and are produced by the slower growth of fatigue cracks at this location in the part.
4. The final fracture surface often looks rough and is in- dicative of brittle fracture, but it can also appear ductile depending on the material.
The actual pattern of striations depends on the particular geometry, material, and loading (Fig. 7.6), and can require experience to evaluate a failure cause.
10-2
10-4
10-6
10-8
Crack growth rate, dlc/dN (mm/cycle)
log ∆K
1 mm/min
1 mm/hr
1 mm/day 1 mm/week Cr
ack growth rate at 50 Hz
= C(∆K)m dlc
dN
Regime A Regime B
Regime C one lattice
spacing per cycle
m 1
Kc
Crack length, lc
Number of cylces, N
∆σ2 > ∆σ1
∆σ2 ∆σ1 dlc
dN
(a)
(b)
Figure 7.4: Illustration of fatigue crack growth. (a) Size of a fatigue crack for two different stress ratios as a function of the number of cycles; (b) rate of crack growth, illustrating three regimes.
Rough (fracture) surface
A B
Striations (visible) Microscopic striations Smooth (burnished) surface
Figure 7.5: Cross section of a fatigued section, showing fa- tigue striations or beachmarks originating from a fatigue crack at B.Source:Rimnac, C., et al., inASTM STP 918, Case Histories Involving Fatigue and Fracture, copyright 1986, ASTM International. Reprinted with permission.
Fatigue Strength 165
High Nominal Stress Low Nominal Stress
No stress
concentration Mild stress
concentration Severe stress
concentration No stress
concentration Mild stress
concentration Severe stress concentration
Tension-tension or tension-compressionUnidirectional bendingReversed bending
Rotational bending
Beachmarks Fracture surface
Figure 7.6: Typical fatigue-fracture surfaces of smooth and notched cross-sections under different loading conditions and stress levels.Source:Adapted fromMetals Handbook, American Society for Metals [1975].
7.5.4 S-N Diagrams
Data from reversed bending experiments are plotted as the fatigue strength versus the logarithm of the total number of cycles to failure,Nt0, for each specimen. These plots are called S-N diagramsorW ¨ohler diagramsafter August W ¨ohler, a German engineer who published his fatigue research in 1870.
They are a standard method of presenting fatigue data and are useful and informative. Two general patterns for two classes of material, those with and those without endurance limits, emerge when plotting the fatigue strength versus the logarithm of the number of cycles to failure. Figure 7.7 shows typical results for several materials. Figure 7.7a presents test data for wrought steel. Note the large amount of scatter in the data, even with the great care used to prepare test spec- imens. Thus, material properties extracted from curves such as those in Fig. 7.7 are all somewhat suspect, and have signifi- cant variation between test specimens. Figure 7.7a also shows a common result. For some materials withendurance limits, such as ferrous and titanium alloys, a change in slope occurs at low stress levels, called a “knee” in the curve. This implies that an endurance limitSe0 is reached, below which failure will not occur (although this is strictly not true — see Sec- tion 7.6.3). This endurance limitS0erepresents the largest fluc- tuating stress that willnotcause failure for an infinite number of cycles. For many steels the endurance limit ranges between 35 and 60% of the ultimate strength.
Most nonferrous alloys (e.g., aluminum, copper, and magnesium)do nothave a significant endurance limit. Their fatigue strength continues to decrease with increasing cycles.
Thus, fatigue will occur regardless of the stress amplitude.
The fatigue strength for these materials is taken as the stress level at which failure will occur for some specified number of cycles (e.g.,106or107cycles).
Determining the endurance limit experimentally is lengthy and expensive. The Manson-Coffin relationship given by Eq. (7.5) demonstrates that the fatigue life will de- pend on the material’s fracture strength during a single load cycle, suggesting a possible relationship between static mate- rial strength and strength in fatigue. Such a relationship has been noted by several researchers (see Fig. 7.8). The stress endurance limits of steel for three types of loading can be ap- proximated as
bending : Se0 = 0.5Su
axial : Se0 = 0.45Su
torsion : Se0 = 0.29Su (7.6) Equation (7.6) can be used to approximate the endurance limits for other ferrous alloys but it must be recognized that the limits can vary significantly from experimentally deter- mined endurance limits. As depicted by the dashed line in Fig. 7.8, the maximum value of the endurance limit for fer- rous alloys is taken as 690 MPa, regardless of the predictions from Eq. (7.6). Even if the ultimate strength and the type of loading are known for other ferrous metals, their endurance limits can be only be approximated from Eq. (7.6).
Other materials, for which there is much less experience, are nevertheless finding increasing uses in fatigue applica- tions. Table 7.2 gives the approximate strengths in fatigue for various material classes. Figure 7.7c gives some stress- life curves for common polymers. Because polymers have a much greater variation in properties than metals, Fig. 7.7c should be viewed as illustrative of fatigue properties and not used for quantitative data.
Given a new material, or when experimental verification of an endurance limit is needed, it is often not required to de- velop the entire S-N diagram. The designer may only wish to
(a)
(c) (b) Alternating stress, σa, MPa
Wrought
Sand cast Permanent mold cast
40 80 160 320
640 Aluminum Alloys
103 104 105 106 107 108 109
1.0 0.9 0.8 0.7 0.6
0.5
0.4
103 104 105 106 107
Number of cycles to failure, N’
Fatigue stress ratio, Sf/Sut
Not broken Ferrous Alloys
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PTFE
Phenolic Epoxy
Diallyl- phthalate Alkyd
Nylon (dry) Polycarbonate
104
103 105 106 107
Number of cycles to failure, N’
Alternating stress, σa, MPa 0 10 20 30 40 50
60 Polymers
Number of cycles to failure, N’
Figure 7.7: Fatigue strength as a function of number of load- ing cycles. (a) Ferrous alloys, showing clear endurance limit;
(b) aluminum alloys, with less pronounced knee and no en- durance limit; (c) selected properties of assorted polymer classes.Source:(a) Adapted from Lipson and Juvinall [1963], (b) Adapted from Juvinall and Marshek [1991], (c) Adapted from Norton [1996].
Fatigue Strength 167
690 MPa 1200
900
600
300
00 2000
S'___e
Su= 0.6 0.5 0.4
Tensile strength, Sut, MPa Endurance limit,S’e, MPa
▲▲▲▲▲▲▲
1000
500 1500 Carbon steels
Alloy steels Wrought irons
▲
Figure 7.8: Endurance limit as a function of ultimate strength for wrought steels. Source: Adapted from Shigley and Mitchell [1983].
experimentally obtain the endurance limit, or the endurance limit at a desired number of stress cycles, using a minimum number of specimens in order to control costs and time re- quired for evaluation. A valuable approach in this case is the staircaseapproach, also known as theup-and-downmethod as outlined in Design Procedure 7.2.