Chapter 7: Fatigue and Impact All 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 anributed to corrosion- assisted fatigue. (Steven Minkowski/Gamma Liaison) Fundamentals of Machine Elements, 3rd ed.
On 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.
Design Procedure 7.1: Methods to Maximize Fatigue Life 1. Minimizing 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. 2. 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. 3. 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). 4. Maximizing the critical crack length. Fracture toughness (Section 6.5) is an essential material property, and materials with higher fracture toughnesses are generally bener suited for fatigue applications.
Cyclic Stresses Mean stress: Stress Tension + Compression max 0 min m 1 cycle Time a r Figure 7.2: Variation in nonzero cyclic mean stress. Stress range: Stress amplitude: Stress ratio: σ m = σ max + σ min 2 σ r = σ max σ min σ a = σ r 2 = σ max σ min 2 R = σ min σ max
Cyclic Properties of Metals Yield Fracture Fatigue Fatigue Fatigue strength strength ductility strength ductility S y, f, coef, exponent, exponent, Material Condition a MPa MPa f a Steel 1015 Normalized 228 827 0.95-0.110-0.64 4340 Tempered 1172 1655 0.73-0.076-0.62 1045 Q&T 306 F 1720 2720 0.07-0.055-0.60 1045 Q&T 500 F 1275 2275 0.25-0.080-0.68 1045 Q&T 600 F 965 1790 0.35-0.070-0.69 4142 Q&T 400 F 1720 2650 0.07-0.076-0.76 4142 Q&T 600 F 1340 2170 0.09-0.081-0.66 4142 Q&T 700 F 1070 2000 0.40-0.080-0.73 4142 Q&T 840 F 900 1550 0.45-0.080-0.75 Aluminum 1100 Annealed 97 193 1.80-0.106-0.69 2014 2024 T6 T351 462 379 848 1103 0.42 0.22-0.106-0.124-0.65-0.59 5456 H311 234 724 0.46-0.110-0.67 7075 Titanium T6 469 1317 0.19-0.126-0.52 Ti-6Al-4V Solution treated+aged 1185 2030 0.841-0.104-0.69 Nickel Inconel X Annealed 700 2255 1.16-0.117-0.75 a Q&T - Quenched and tempered. Table 7.1: Cyclic properties of some metals. Source: After Shigley and Mitchell [1983] and Suresh [1998].
Common Stress PaNerns and R.R. Moore Test Specimen Four frequently encountered panerns of constant- amplitude cyclic stress are: 1. Completely reversed: (σ m = 0, R = - 1) 2. Nonzero mean: (as shown in Fig. 7.2) 3. Released tension: (σ min = 0, R = 0, σ m = σ max /2) 4. Released compression: (σ max = 0, R =, σ m = σ min /2. 0.30 3 7 16 9 7 8 R Figure 7.3: R.R. Moore machine fatigue test specimen. Dimensions in inches.
Fatigue Crack Growth Crack length, l c 2 > 1 2 1 dl c dn Number of cylces, N Crack growth rate, dl c /dn (mm/cycle) 10-2 10-4 10-6 10-8 Regime A one lattice spacing per cycle Regime B m 1 log K K c dl c = C(K) m dn Regime C 1 mm/min 1 mm/hr 1 mm/day 1 mm/week Crack growth rate at 50 Hz (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.
Fatigue Crack Growth Notes Strain- life theory (Manson- Coffin relationship): 2 = σ f E (2N ) a + f (2N ) α Regimes of Crack Growth: 1. 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. 2. Regime B is a period of moderate crack growth rate, often referred to as the Paris regime 3. Regime 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.
Fatigue Striations Smooth (burnished) surface Microscopic striations B Striations (visible) Rough (fracture) surface A Figure 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.
No stress concentration High Nominal Stress Mild stress concentration Severe stress concentration No stress concentration Low Nominal Stress Mild stress concentration Severe stress concentration Fatigue Tension-tension or tension-compression Fracture Surfaces Reversed bending Unidirectional 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: Metals Handbook, American Society for Metals [1975].
Fatigue Strength Ferrous Alloys Fatigue stress ratio, S f /S ut 1.0 0.9 0.8 0.7 0.6 0.5 Ferrous Alloys Not broken For steels: bending : axial : torsion : Se =0.5S u Se =0.45S u Se =0.29S u 0.4 10 3 10 4 10 5 10 6 10 7 Number of cycles to failure, N (a) Figure 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].
Fatigue Strength Nonferrous Alloys 80 60 Aluminum Alloys Alternating stress, a, ksi 40 30 20 16 12 10 Wrought Permanent mold cast Sand cast 8 7 6 5 10 3 10 4 10 5 10 6 10 7 10 8 10 9 Number of cycles to failure, N (b) 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].
Fatigue Strength - Polymers Alternating stress, a, MPa 60 50 40 30 20 10 Alkyd PTFE Polymers Phenolic Epoxy Diallylphthalate Nylon (dry) Polycarbonate 8x10 3 6 4 2 Alternating stress, a, psi 0 0 10 3 10 4 10 5 10 6 10 7 Number of cycles to failure, N (c) 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]
Endurance Limit vs. Ultimate Strength 160 Endurance limit,, ksi 120 80 40 Carbon steels Alloy steels Wrought irons S' e S = 0.6 u 0.5 0.4 100 ksi 0 0 60 120 180 240 300 Tensile strength, S ut, ksi Figure 7.8: Endurance limit as function of ultimate strength for wrought steels. Source: Adapted from Shigley and Mitchell [1983].
Staircase Approach Applied stress, MPa 475 450 425 400 375 350 0 Test number Failure Survival 5 10 15 20 25 Figure 7.9: Typical results from fatigue tests using the staircase approach, and used in Example 7.2. Design Procedure 7.2: Staircase Approach 1. 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. 2. A test interval is then selected, typically around 10% of the estimated endurance limit. 3. An initial test is performed at a stress level equal to the expected endurance limit. 4. 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.
Design Procedure 7.2 (concluded) 5. At the desired duration (commonly 10 6 or 10 7 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. 6. A plot of typical results is shown in Fig. 7.9. 7. The mean endurance limit can be obtained from the following steps: a. Count the number of failures and survivals in the test results. Proceed with the analysis using the less common test result. b. The number of events (failures or survivals) is assigned to n i for each stress level σ i. In this approach, the lowest stress level is denoted as σ o, the next highest as σ 1, etc. c. Obtain the quantity A n from A n = in i d. The endurance limit is then estimated from S e = σ o + d An ± 1 ni 2 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. 8. It is recommended that at least 15 experiments be performed, although more can be helpful for more accurate quantification of the endurance limit.
Endurance Limit for Materials Number of Material cycles Relation Magnesium alloys 10 8 = 0.35S u Copper alloys 10 8 0.25S u < < 0.5S u Nickel alloys 10 8 0.35S u < < 0.5S u Titanium 10 7 0.45S u < < 0.65S u Aluminum alloys 5 10 8 = 0.40S u (S u < 48 ksi) = 19 ksi (S u 48 ksi) Table 7.2: Approximate endurance limit for various materials. Source: Adapted from Juvinall and Marshek [1991].
Finite Life Fatigue Low cycle (below around 1000 cycles): bending: axial: torsion: Sl =0.9S u Sl =0.75S u Sl =0.72S u High cycle, finite life (between around 1000 and 1 million cycles) log S f = b s log N t + C where b s = 1 3 log S l S e S C = 2 log l Se + log Se = log (Sl )2 Se.
Notch Sensitivity Use these values with bending and axial loads Use these values with torsion Notch sensitivity, q n 1.0 0.8 0.6 0.4 200 (1379) 140 (965) 100 (689) 80 (552) 60 (414) 50 (345) 180 (1241) 120 (827) 80 (552) 60 (414) Aluminum alloy (based on 2024-T6 data) Steel, S u, ksi (MPa) as marked Fatigue stress concentration factor: K f =1+(K c 1) q n 0.2 0 0 0.5 1.0 1.5 2.0 Notch radius, r, mm 2.5 3.0 3.5 4.0 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Notch radius, r, in. Figure 7.10: Notch sensitivity as function of notch radius for several materials and types of loading. Source: Adapted from Sines and Waisman [1959].
Modified Endurance Limit The modified endurance limit can be estimated from an R.R. Moore idealized specimen from: = k f k s k r k t k m S e This is strictly true only for carbon steels. Correction factors can be estimated from empirical relations. Experimental verification of designs is usually required.
Surface Finish Correction Mathematical estimate: k f = es f ut Note: not based on curve fit of Fig. 7.11. Manufacturing Factor e process MPa ksi Exponent f Grinding 1.58 1.34-0.085 Machining or 4.51 2.70-0.265 cold drawing Hot rolling 57.7 14.4-0.718 As forged 272.0 39.9-0.995 Table 7.3: Surface finish factor. Source: Shigley and Mitchell [1983].
Surface Finish Correction Surface finish factor, k f 1.0 0.8 0.6 0.4 0.2 Fine polishing Tap water corroded Machined, cold forged, cold rolled Hot rolled (a) Hot forged Salt water corroded 0 60 100 140 180 220 260 Tensile strength, S ut (ksi) Figure 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]. Surface finish factor, k f 1.0 0.9 0.8 0.7 0.6 0.5 Surface finish R a, in. 2000 1000 500 0.4 40 80 120 160 200 240 (b) 125 250 16 32 63 Ultimate strength in tension, S ut, ksi 1 8 4 2
Reliability, Size and Temperature Factor Reliability Factor: For a standard deviation of 8% of the mean: k r =0.512 ln Probability Reliability of survival, factor, percent k r 50 1.00 90 0.90 95 0.87 99 0.82 99.9 0.75 99.99 0.70 0.11 1 +0.508 R Size Factor: 0.869d 0.112 0.3 in.<d<10 in. k s = 1 d<0.3 in. or d 8 mm 1.248d 0.112 8 mm <d 250 mm d depends on manufacturing process, but one approach allows estimation from the equivalent area where the stress is above 95% of the maximum stress: A95 d = 0.0766 Table 7.4: Reliability factors for six probabilities of survival. Temperature Factor: k t = S ut S ut,ref
Shot Peening Effect Fatigue strength at two million cycles (MPa) 1380 1035 690 345 0 100 200 300 Peened - smooth or notched ksi Not peened - smooth Not peened - notched (typical machined surface) 690 1380 2170 Ultimate tensile strength, S ut, (MPa) 200 150 100 50 0 ksi Alternating stress, a, MPa 483 414 345 276 207 138 Machined Shot peened Al 7050-T7651 Ti-6Al-4V Polished 10 4 10 5 10 6 10 7 10 8 Number of cycles to failure, N' 70 60 50 40 30 20 ksi (a) Figure 7.12: The use of shot peening to improve fatigue properties. (a) Fatigue strength at 2 x 10 6 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. (b)
Design Procedure 7.3: Determination of Endurance Limit If an experimental investigation is impractical, the endurance limit can be estimated through the following procedure: 1. The endurance limit for a specimen ( ) can be estimated for a type of loading from Eq. (7.6). This requires knowledge of the material'ʹs ultimate strength, which can be obtained from experiments or from tables of mechanical properties; some steel properties are summarized in Appendix A. 2. Note from Fig. 7.8 that the predicted value should not be assigned a value greater than 690 MPa (100 ksi). 3. The modified endurance limit ( ) is then obtained from Eq. (7.18), where: a. The surface finish factor, k f, is obtained from Eq. (7.19) using coefficients from Table 7.3, or else k f can be estimated from Fig. 7.11. b. The size factor, k s, can be estimated from Eq. (7.20) for bending or torsion, with k s =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. c. The reliability factor, k r, can be obtained from Table 7.4. d. The effects of temperature, k t, are best obtained experimentally, but Eq.~(7.23) gives a reasonable estimate for this factor.
Examples 7.5 and 7.6 r=0.2 r = 2 M 2 2 M M 45 45 41 M 50 Figure 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.
Haigh Diagram a S ut m S ut max /S ut 1.0 0.8 0.6 0.4 R = -1.0 0.8 0.6 0.4 10 4 cycles 10 5 cycles 10 6 cycles R = -0.5 R = 0.0 R = 0.5 0.4 R = 1.0 0.6 0.8 0.2 0.2 0.2 0.0-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 min /S ut Figure 7.15: A typical Haigh diagram showing constant life curves for different combinations of mean and alternating stresses.
Nonzero Mean Stress S yt Yield line Gerber: Alternating stress, a 0 Soderberg line Gerber line Mean stress, m Goodman line Figure 7.16: Influence of nonzero mean stress on fatigue life for tensile loading as estimated by four empirical relationships. S yt S ut K f n s σ a + Goodman: K f σ a Soderberg: K f σ a 2 ns σ m =1 S ut + σ m S ut = 1 n s + σ m S yt = 1 n s
Modified Goodman Equations Line Equation AB σ max = K f + σ m 1 BC σ max = S y S y Range 0 σ m S y /K f S u K f 1 K f S u 1 CD σ min = 2 σ m S y S y DE σ min = 1 + EF σ min = σ m K f 1 K f K f S u K f K f S u K f S u σ m K f 0 σ m σ m σ m S y S y S y K f 1 K f S u K f S y σ m 0 FG σ min = S y S y σ m K f GH σ max = 2 σ m + S y S y σ m HA σ max = σ m + K f K f K f S y σ m 0 S y S y Table 7.5: Equations and range of applicability for construction of complete modified Goodman diagram.
Modified Goodman Diagram S u S u + S y N B C S y max max H /K f 45 A L M D m min m 0 S y S u m min E /K f G F 45 S y a b c d Figure 7.17: Complete modified Goodman diagram, ploning stress as ordinate and mean stress as abscissa.
Modified Goodman Equations Region in Failure Validity limits Fig. 7.16 equation of equation a σ max 2σ m = S y /n s S y σ m b σ max σ m = n s K f c σ max + σ m K f S u 1 = d σ max = S y n s K f K f S y σ m 0 n s K f 0 σ m S y 1 K f K f S u S y 1 σ m S y K f K f S u S y Table 7.6: Failure equations and validity limits of equations for four regions of complete modified Goodman relationship
Alternating Stress Ratio Alternating stress ratio, a /S u 1.5 1.0 0.5 0 4.0 S e (0.4)(0.9) = 0.36 S u 3.0 2.0 1.0 0 1.0 Mean stress ratio, m /S u Figure 7.18: Alternating stress ratio as function of mean stress ratio for axially loaded cast iron.
Fatigue Crack Growth Data Rate of crack growth, dlc/dn (mm/cycle) 10-1 10-2 10-3 10-4 10-5 10-6 PMMA Epoxy PC PSF Nylon 66 Nylon ST 801 PVC PET Al 2219-T851 300M Steel Rate of crack growth, dlc/dn (mm/cycle) 10-4 10-5 10-6 10-7 Mg 7075-T6 Al A36 steel Ti-6Al-4V 2024-T3 Al Mo 4340 steel 0.2 0.4 0.8 1.0 2 4 8 10 20 40 80 K (MPa m) 10-8 1 2 5 10 20 50 100 K (MPa m) (a) Figure 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]. (b)
Paris Law Data Material C mm/cycle in/cycle ( MPa m) m ksi in m Steel Ferritic- pearlitic 6.89 10 9 3.6 10 10 3.0 Martensitic 1.36 10 7 6.6 10 9 2.25 Austenitic 5.61 10 9 3.0 10 10 3.25 Aluminum 6061- T6 5.88 10 8 3.1 10 9 3.17 2024- T3 1.6 10 11 8.4 10 11 3.59 m Paris law: dl c dn = C ( K)m Table 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].
Dynamic Mechanical Properties Ultimate and yield stresses, S u and S y, ksi 100 80 60 40 20 Ratio S y /S u Yield strength S y Ultimate strength S u Total elongation 0 10 6 10 4 10 2 1 10 2 Average strain rate, s 1 100 80 60 40 20 0 Elongation, percent S y /S u, percent Figure 7.20: Mechanical properties of mild steel at room temperature as function of average strain rate.
Example 7.11 V y 1.5 m (a) 0.6 m 40 mm 450 mm (b) M x (c) P Figure 7.21: Diver impacting diving board, used in Example 7.11. (a) Side view; (b) front view; (c) side view showing forces and coordinates.
D- Check (a) (b) Figure 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.