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Chapter 7: Fatigue and ImpactAll machine and structural designs are problems in fatigue because the forces of Nature are always at work and each object must respond in some fashion.

Carl Osgood, Fatigue Design Aloha Airlines Flight 243, a Boeing 737-200, taken April 28, 1988. The midflight fuselage failure was attributed to corrosion-assisted fatigue. (Steven Minkowski/Gamma Liaison)

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressOn the Bridge!Figure 7.1: On the Bridge, an illustration from Punch magazine in 1891 warning the populace that death was waiting for them on the next bridge. Note the cracks in the iron bridge.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressDesign Procedure 7.1: Methods to Maximize Fatigue LifeMinimizing initial flaws, especially surface flaws. Great care is taken to produce fatigue-resistant surfaces through processes such as grinding or polishing that produce exceptionally smooth surfaces. These surfaces are then carefully protected before a product is placed into service.Maximizing crack initiation time. Compressive surface residual stresses are imparted (or at least tensile residual stresses are relieved) through manufacturing processes such as shot peening or burnishing, or by a number of surface treatments. Maximizing crack propagation time. Substrate properties, especially those that retard crack growth, are also important. For example, in some materials fatigue cracks will propagate more quickly along grain boundaries than through grains. In this case, using a material that has elongated grains transverse to the direction of fatigue crack growth can extend fatigue life (e.g., by using cold-worked components instead of castings). Maximizing the critical crack length. Fracture toughness (Section 6.5) is an essential material property, and materials with higher fracture toughnesses are generally better suited for fatigue applications.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressCyclic StressesFigure 7.2: Variation in nonzero cyclic mean stress.

Mean stress:Stress range:Stress amplitude:Stress ratio:

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressCyclic Properties of MetalsTable 7.1: Cyclic properties of some metals. Source: After Shigley and Mitchell [1983] and Suresh [1998].

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressCommon Stress Patterns and R.R. Moore Test SpecimenFigure 7.3: R.R. Moore machine fatigue test specimen. Dimensions in inches.

Four frequently encountered patterns of constant-amplitude cyclic stress are:Completely reversed: (m = 0, R = -1) Nonzero mean: (as shown in Fig. 7.2)Released tension: (min = 0, R = 0, m = max/2)Released compression: (max= 0, R = , m = min/2.Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressFatigue Crack GrowthFigure 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.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressFatigue Crack Growth Notes

Strain-life theory (Manson-Coffin relationship):Regimes of Crack Growth:Regime A is 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 B is a period of moderate crack growth rate, often referred to as the Paris regimeRegime C is a period of high-growth rate, where the maximum stress intensity factor for the fatigue cycle approaches the fracture toughness of the material. Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressFatigue StriationsFigure 7.5: Cross section of a fatigued section, showing fatigue striations or beachmarks originating from a fatigue crack at B. Source: Rimnac, C., et al., in ASTM STP 918, Case Histories Involving Fatigue and Fracture, copyright 1986, ASTM International. Reprinted with permission.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressFatigue Fracture SurfacesFigure 7.6: Typical fatigue-fracture surfaces of smooth and notched cross-sections under different loading conditions and stress levels. Source: Metals Handbook, American Society for Metals [1975].

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressFatigue Strength Ferrous AlloysFigure 7.7: Fatigue strength as a function of number of loading cycles. (a) Ferrous alloys, showing clear endurance limit; Source: Adapted from Lipson and Juvinall [1963].

For steels:

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressFatigue Strength Nonferrous Alloys

Figure 7.7: Fatigue strength as a function of number of loading cycles. (b) aluminum alloys, with less pronounced knee and no endurance limit. Source: Adapted from Juvinall and Marshek [1991].Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressFatigue Strength - Polymers

Figure 7.7: Fatigue strength as a function of number of loading cycles. (c) selected properties of assorted polymer classes. Source: Adapted from Norton [1996]Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressEndurance Limit vs. Ultimate StrengthFigure 7.8: Endurance limit as function of ultimate strength for wrought steels. Source: Adapted from Shigley and Mitchell [1983].

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressStaircase ApproachFigure 7.9: Typical results from fatigue tests using the staircase approach, and used in Example 7.2.

Design Procedure 7.2: Staircase Approach

A designer must first estimate the endurance limit for the material of interest, either with a strength-based approach such as in Eq. (7.6), or through preliminary testing.A test interval is then selected, typically around 10% of the estimated endurance limit. An initial test is performed at a stress level equal to the expected endurance limit. If the specimen breaks, it is recorded as such and the next experiment will be performed at a stress level reduced by the stress interval. Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressDesign Procedure 7.2 (concluded)At the desired duration (commonly 106 or 107 cycles), the test is stopped. If the specimen survives, it is recorded as such and the next experiment will be performed at a stress level increased by the stress interval. A plot of typical results is shown in Fig. 7.9.The mean endurance limit can be obtained from the following steps:Count the number of failures and survivals in the test results. Proceed with the analysis using the less common test result. The number of events (failures or survivals) is assigned to ni for each stress level i. In this approach, the lowest stress level is denoted as o, the next highest as 1, etc.Obtain the quantity An from

The endurance limit is then estimated from

where the plus sign is used if the more common experimental result is survival, and the minus sign is used if the more common event is failure.It is recommended that at least 15 experiments be performed, although more can be helpful for more accurate quantification of the endurance limit.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressEndurance Limit for MaterialsTable 7.2: Approximate endurance limit for various materials. Source: Adapted from Juvinall and Marshek [1991].

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressFinite Life FatigueLow cycle (below around 1000 cycles):

High cycle, finite life (between around 1000 and 1 million cycles)

whereFundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressNotch SensitivityFigure 7.10: Notch sensitivity as function of notch radius for several materials and types of loading. Source: Adapted from Sines and Waisman [1959].

Fatigue stress concentration factor:Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressThe modified endurance limit can be estimated from an R.R. Moore idealized specimen from:

This is strictly true only for carbon steels.

Correction factors can be estimated from empirical relations. Experimental verification of designs is usually required.Modified Endurance Limit

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressMathematical estimate:

Note: not based on curve fit of Fig. 7.11.Surface Finish CorrectionTable 7.3: Surface finish factor. Source: Shigley and Mitchell [1983].

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressSurface Finish CorrectionFigure 7.11: Surface finish factors for steel. (a) As function of ultimate strength in tension for different manufacturing processes; Source: (a) Adapted from Norton [2011] and data from the American Iron and Steel Institute. (b) As function of ultimate strength and surface roughness as measured with a stylus profilometer. Source: (b) adapted from Johnson [1967].

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressReliability, Size and Temperature FactorTable 7.4: Reliability factors for six probabilities of survival.

Reliability Factor:For a standard deviation of 8% of the mean: Size Factor:

d depends on manufacturing process, but one approach allows estimation from the equivalent area where the stress is above 95% of the maximum stress:

Temperature Factor:

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressShot Peening EffectFigure 7.12: The use of shot peening to improve fatigue properties. (a) Fatigue strength at 2 x 106 cycles for high-strength steel as a function of ultimate strength; (b) typical S-N curves for non-ferrous metals. Source: Courtesy of J.~Champaigne, Electronics, Inc.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressDesign Procedure 7.3:Determination of Endurance LimitIf an experimental investigation is impractical, the endurance limit can be estimated through the following procedure:The endurance limit for a specimen (Se) can be estimated for a type of loading from Eq. (7.6). This requires knowledge of the material's ultimate strength, which can be obtained from experiments or from tables of mechanical properties; some steel properties are summarized in Appendix A. Note from Fig. 7.8 that the predicted value should not be assigned a value greater than 690 MPa (100 ksi). The modified endurance limit (Se) is then obtained from Eq. (7.18), where:The surface finish factor, kf, is obtained from Eq. (7.19) using coefficients from Table 7.3, or else kf can be estimated from Fig. 7.11.The size factor, ks, can be estimated from Eq. (7.20) for bending or torsion, with ks=1 for tension. If the part is not round, then an equivalent diameter can be obtained from Eq. (7.21). These equations have high uncertainty, but they do allow size effects to be considered without overly complicating the mathematics.The reliability factor, kr, can be obtained from Table 7.4.The effects of temperature, kt, are best obtained experimentally, but Eq.~(7.23) gives a reasonable estimate for this factor.Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressExamples 7.5 and 7.6Figure 7.13: Round shaft with a retaining ring groove considered in Example 7.5. All dimensions are in millimeters.

Figure 7.14: Drawn square profile with machined groove considered in Example 7.6. All dimensions are in millimeters.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressHaigh DiagramFigure 7.15: A typical Haigh diagram showing constant life curves for different combinations of mean and alternating stresses.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressNonzero Mean StressFigure 7.16: Influence of nonzero mean stress on fatigue life for tensile loading as estimated by four empirical relationships.

Gerber:Goodman:Soderberg:

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressModified Goodman EquationsTable 7.5: Equations and range of applicability for construction of complete modified Goodman diagram.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressModified Goodman DiagramFigure 7.17: Complete modified Goodman diagram, plotting stress as ordinate and mean stress as abscissa.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressModified Goodman EquationsTable 7.6: Failure equations and validity limits of equations for four regions of complete modified Goodman relationship

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressAlternating Stress RatioFigure 7.18: Alternating stress ratio as function of mean stress ratio for axially loaded cast iron.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressFatigue Crack Growth DataFigure 7.19: Fatigue crack growth data for a variety of materials. (a) Selected polymers in comparison to aluminum and steel; (b) selected metal alloys. {\it Source:} From Bowman [2004].

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressParis Law DataTable 7.6: Paris Law constants for various classes of steel. Data represents worst-case (fastest) crack growth rates reported for the material classes. Source: From Dowling [2007].

Paris law:Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressDynamic Mechanical PropertiesFigure 7.20: Mechanical properties of mild steel at room temperature as function of average strain rate.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressExample 7.11Figure 7.21: Diver impacting diving board, used in Example 7.11. (a) Side view; (b) front view; (c) side view showing forces and coordinates.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC PressD-CheckFigure 7.22: (a) Exterior view of Boeing 747-400 during a D check; (b) inspection of landing gear component for structural integrity. Source: Courtesy of Lufthansa Technik.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson 2014 CRC Press