Multiaxial Fatigue. Professor Darrell Socie Darrell Socie, All Rights Reserved
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1 Multiaxial Fatigue Professor Darrell Socie Darrell Socie, All Rights Reserved
2 Outline Stresses around holes Crack Nucleation Crack Growth MultiaxialFatigue Darrell Socie, All Rights Reserved 1 of 85
3 Multiaxial Fatigue Problems Uniaxial loading that produces multi stressesund stress concentrators Multiaxial loading that produces uniaxial stresses around stress concentrators Multiaxial loading that produces multiaxial stresses around stress concentrators Multiaxial loading that causes mixed mode long crack growth MultiaxialFatigue Darrell Socie, All Rights Reserved 2 of 85
4 3D stresses Longitudinal Tensile Strain Transverse Compression Strain Thickness 50 mm 30 mm 15 mm 7 mm y z x 100 MultiaxialFatigue Darrell Socie, All Rights Reserved 3 of 85
5 Notch Stresses y z x t x z x z MultiaxialFatigue Darrell Socie, All Rights Reserved 4 of 85
6 Multiaxial Fatigue Problems Uniaxial loading that produces multi stressesund stress concentrators Multiaxial loading that produces uniaxial stresses around stress concentrators Multiaxial loading that produces multiaxial stresses around stress concentrators Multiaxial loading that causes mixed mode long crack growth MultiaxialFatigue Darrell Socie, All Rights Reserved 5 of 85
7 Hole in a Plate a r r r r r MultiaxialFatigue Darrell Socie, All Rights Reserved 6 of 85
8 Stresses at the Hole = Angle = -1 = 0 Stress concentration factor depends on type of loading MultiaxialFatigue Darrell Socie, All Rights Reserved 7 of 85
9 Combined Loading = MultiaxialFatigue Darrell Socie, All Rights Reserved 8 of 85
10 Maximum Tensile Stress Location K t = 3 1 = K t = = 1.72 K t = 4 1 = MultiaxialFatigue Darrell Socie, All Rights Reserved 9 of 85
11 In and Out of Phase Loading In-phase Out-of-phase K t = 3 K t = 4 Damage location changes with load phasing MultiaxialFatigue Darrell Socie, All Rights Reserved 10 of 85
12 Multiaxial Fatigue Problems Uniaxial loading that produces multiaxial stresses around stress concentrators Multiaxial loading that produces uniaxial stresses around stress concentrators Multiaxial loading that produces multiaxial stresses around stress concentrators Multiaxial loading that causes mixed mode long crack growth MultiaxialFatigue Darrell Socie, All Rights Reserved 11 of 85
13 Notched Shaft Loading M T M X P M Y z z MultiaxialFatigue Darrell Socie, All Rights Reserved 12 of 85
14 Torsion Loading M X t 1 T 1 t 3 M T z z M X T T T t 2 t 4 M T z t 1 t 2 t 3 t 4 T 1 T Out-of-phase shear loading is needed to produce nonproportional stressing MultiaxialFatigue Darrell Socie, All Rights Reserved 13 of 85
15 Multiaxial Fatigue Problems Uniaxial loading that produces multiaxial stresses around stress concentrators Multiaxial loading that produces uniaxial stresses around stress concentrators Multiaxial loading that produces multiaxial stresses around stress concentrators Multiaxial loading that causes mixed mode long crack growth MultiaxialFatigue Darrell Socie, All Rights Reserved 14 of 85
16 Shear Stresses Around Hole = = 0 r xy r a MultiaxialFatigue Darrell Socie, All Rights Reserved 15 of 85
17 Torsion Experiments MultiaxialFatigue Darrell Socie, All Rights Reserved 16 of 85
18 Shear Stresses Around Hole r = 1.33 = 0 r xy r 1.33 a Angle MultiaxialFatigue Darrell Socie, All Rights Reserved 17 of 85
19 Stress Intensity Factors F R 0.50 a a R da dn C K m eq K I F a MultiaxialFatigue Darrell Socie, All Rights Reserved 18 of 85
20 Cracks from Holes Crack nucleates in shear Mixed mode growth? Tensile mode growth? MultiaxialFatigue Darrell Socie, All Rights Reserved 19 of 85
21 Summary Cracks nucleate in a uniaxial stress field and then grow in a mixed tensile/shear stress field MultiaxialFatigue Darrell Socie, All Rights Reserved 20 of 85
22 Outline Stresses around holes Crack Nucleation Stress Based Models Strain Based Models Crack Growth MultiaxialFatigue Darrell Socie, All Rights Reserved 21 of 85
23 Fatigue Mechanisms Summary Fatigue cracks nucleate in shear Fatigue cracks grow in either shear or tension depending on material and state of stress MultiaxialFatigue Darrell Socie, All Rights Reserved 22 of 85
24 Stress Based Models Sines Findley Dang Van MultiaxialFatigue Darrell Socie, All Rights Reserved 23 of 85
25 Bending Torsion Correlation Shear stress in bending 1/2 Bending fatigue limit Shear stress Octahedral stress Principal stress Shear stress in torsion 1/2 Bending fatigue limit MultiaxialFatigue Darrell Socie, All Rights Reserved 24 of 85
26 Test Results Cyclic tension with static tension Cyclic torsion with static torsion Cyclic tension with static torsion Cyclic torsion with static tension MultiaxialFatigue Darrell Socie, All Rights Reserved 25 of 85
27 Cyclic Tension with Static Tension 1.5 Axial stress Fatigue strength Mean stress Yield strength 1.5 MultiaxialFatigue Darrell Socie, All Rights Reserved 26 of 85
28 Cyclic Torsion with Static Torsion 1.5 Shear Stress Amplitude Shear Fatigue Strength Maximum Shear Stress Shear Yield Strength MultiaxialFatigue Darrell Socie, All Rights Reserved 27 of
29 Cyclic Tension with Static Torsion 1.5 Bending Stress Bending Fatigue Strength Static Torsion Stress Torsion Yield Strength 3.0 MultiaxialFatigue Darrell Socie, All Rights Reserved 28 of 85
30 Cyclic Torsion with Static Tension 1.5 Torsion shear stress Shear fatigue strength Axial mean stress 1.5 Yield strength MultiaxialFatigue Darrell Socie, All Rights Reserved 29 of 85
31 Conclusions Tension mean stress affects both tension and torsion Torsion mean stress does not affect tension or torsion MultiaxialFatigue Darrell Socie, All Rights Reserved 30 of 85
32 Sines 2 oct (3 h ) 1 6 ( x ( mean x y ) 2 ( mean y x z mean z ) 2 ) ( y z ) 2 6( 2 xy 2 xz 2 yz ) MultiaxialFatigue Darrell Socie, All Rights Reserved 31 of 85
33 Findley 2 k n max f tension torsion MultiaxialFatigue Darrell Socie, All Rights Reserved 32 of 85
34 Bending Torsion Correlation Shear stress in bending 1/2 Bending fatigue limit Shear stress Octahedral stress Principal stress Shear stress in torsion 1/2 Bending fatigue limit MultiaxialFatigue Darrell Socie, All Rights Reserved 33 of 85
35 Dang Van ij (M,t) E ij (M,t) () t a () t b h m ij (m,t) ij (m,t) V(M) MultiaxialFatigue Darrell Socie, All Rights Reserved 34 of 85
36 Isotropic Hardening stabilized stress range Failure occurs when the stress range is not elastic MultiaxialFatigue Darrell Socie, All Rights Reserved 35 of 85
37 Multiaxial Kinematic and Isotropic a) b) 3 3 Yield domain expands and translates C o O o O o 1 R o c) d) Loading path 3 3 C L R L O L O o O o * stabilized residual stress MultiaxialFatigue Darrell Socie, All Rights Reserved 36 of 85
38 Dang Van ( continued ) ta h t= b Failure predicted h h Loading path MultiaxialFatigue Darrell Socie, All Rights Reserved 37 of 85
39 Stress Based Models Summary Sines: Findley: 2 oct 2 (3 h ) k n max f Dang Van: () t a () t b h MultiaxialFatigue Darrell Socie, All Rights Reserved 38 of 85
40 Model Comparison R = Relative Fatigue Life Goodman Findley Sines Torsion Tension Biaxial Tension Stress ratio, MultiaxialFatigue Darrell Socie, All Rights Reserved 39 of 85
41 Outline Stresses around holes Crack Nucleation Stress Based Models Strain Based Models Crack Growth MultiaxialFatigue Darrell Socie, All Rights Reserved 40 of 85
42 Strain Based Models Brown and Miller Fatemi and Socie Smith Watson and Topper MultiaxialFatigue Darrell Socie, All Rights Reserved 41 of 85
43 Brown and Miller Fatigue Life, Cycles 2 x x x = Normal Strain Amplitude, n MultiaxialFatigue Darrell Socie, All Rights Reserved 42 of 85
44 Case A and B Growth along the surface Growth into the surface MultiaxialFatigue Darrell Socie, All Rights Reserved 43 of 85
45 Brown and Miller ( continued ) Uniaxial Equibiaxial MultiaxialFatigue Darrell Socie, All Rights Reserved 44 of 85
46 Brown and Miller ( continued ) 1 max S n max 2 S n A ' f 2 E n, mean b ' f f ( 2N ) B ( 2N ) f c MultiaxialFatigue Darrell Socie, All Rights Reserved 45 of 85
47 Mohr s Circle max n x ˆ max S 1 n Brown and Miller - Cracks should be equally likely on two planes 90 apart y max n MultiaxialFatigue Darrell Socie, All Rights Reserved 46 of 85
48 / 3 Loading Histories C F G H I J MultiaxialFatigue Darrell Socie, All Rights Reserved 47 of 85
49 Crack Directions MultiaxialFatigue Darrell Socie, All Rights Reserved 48 of 85
50 Hypothesis Fatigue damage is planar in nature The material finds a critical plane for microcrack growth. % of applied stress Stresses are nearly the same over a 10 range of angles MultiaxialFatigue Darrell Socie, All Rights Reserved 49 of 85
51 Stresses on the Planes Shear Tension MultiaxialFatigue Darrell Socie, All Rights Reserved 50 of 85
52 Fatemi and Socie MultiaxialFatigue Darrell Socie, All Rights Reserved 51 of 85
53 Fatigue Lives 12,899 12, ,500 5, ,117 16,062 3 C F G ,676 8,020 7,419 9,139 H I J 5,149 3,488 MultiaxialFatigue Darrell Socie, All Rights Reserved 52 of 85
54 Crack Length Observations 2.5 F-495 H-491 Crack Length, mm J-603 I-471 C-399 G Cycles MultiaxialFatigue Darrell Socie, All Rights Reserved 53 of 85
55 Fatemi and Socie 1 k 2 n,max y ' f (2N f ) G bo ' f (2N ) f co MultiaxialFatigue Darrell Socie, All Rights Reserved 54 of 85
56 Torsion Tests = xy MultiaxialFatigue Darrell Socie, All Rights Reserved 55 of 85
57 304 Stainless Steel MultiaxialFatigue Darrell Socie, All Rights Reserved 56 of 85
58 Smith Watson Topper MultiaxialFatigue Darrell Socie, All Rights Reserved 57 of 85
59 SWT ' 2 1 f 2b ' ' bc n (2N f ) ff(2nf ) 2 E MultiaxialFatigue Darrell Socie, All Rights Reserved 58 of 85
60 Loading Histories MultiaxialFatigue Darrell Socie, All Rights Reserved 59 of 85
61 Stress-Strain Response Case Case Case 3 Shear Stress ( MPa ) Case 4 Case 5 Case MultiaxialFatigue Darrell Socie, All Rights Reserved 60 of 85
62 Maximum Stress 2000 All tests have the same strain ranges Equivalent Stress, MPa Fatigue Life, N f Nonproportional hardening results in lower fatigue lives MultiaxialFatigue Darrell Socie, All Rights Reserved 61 of 85
63 Summary Cracks nucleate in shear and then grow in either shear or tension depending on the material and state of stress MultiaxialFatigue Darrell Socie, All Rights Reserved 62 of 85
64 Separate Tensile and Shear Models Cyclic shear strains Cyclic tensile strains Normal stresses open and close microcracks MultiaxialFatigue Darrell Socie, All Rights Reserved 63 of 85
65 Shear Growth shear stress slip bands 10 m crack growth direction ( From Murakami ) MultiaxialFatigue Darrell Socie, All Rights Reserved 64 of 85
66 Tensile Growth crack growth direction 5 m MultiaxialFatigue Darrell Socie, All Rights Reserved 65 of 85
67 Cyclic Torsion Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Shear Damage Tensile Damage MultiaxialFatigue Darrell Socie, All Rights Reserved 66 of 85
68 Cyclic Torsion with Static Tension Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Tension Shear Damage Tensile Damage MultiaxialFatigue Darrell Socie, All Rights Reserved 67 of 85
69 Cyclic Torsion with Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Shear Damage Tensile Damage MultiaxialFatigue Darrell Socie, All Rights Reserved 68 of 85
70 Cyclic Torsion with Tension and Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Hoop Tension Shear Damage Tensile Damage MultiaxialFatigue Darrell Socie, All Rights Reserved 69 of 85
71 Test Results Load Case /2 hoop MPa axial MPa N f Torsion ,200 with tension ,300 with compression ,000 with tension and compression ,200 MultiaxialFatigue Darrell Socie, All Rights Reserved 70 of 85
72 Conclusions All critical plane models correctly predict these results Hydrostatic stress models can not predict these results MultiaxialFatigue Darrell Socie, All Rights Reserved 71 of 85
73 Loading History Shear strain Axial strain MultiaxialFatigue Darrell Socie, All Rights Reserved 72 of 85
74 Model Comparison Summary of calculated fatigue lives Model Equation Life Epsilon ,060 Garud 6.7 5,210 Ellyin ,450 Brown-Miller ,980 SWT ,930 Liu I ,280 Liu II ,420 Chu ,040 Gamma 26,775 Fatemi-Socie ,350 Glinka ,220 MultiaxialFatigue Darrell Socie, All Rights Reserved 73 of 85
75 Outline Stresses around holes Crack Nucleation Crack Growth MultiaxialFatigue Darrell Socie, All Rights Reserved 74 of 85
76 Mode I and Mode II Surface Cracks Mode II Mode I MultiaxialFatigue Darrell Socie, All Rights Reserved 75 of 85
77 Biaxial Mode I Growth 10-3 = 193 MPa = -1 0 = = 386 MPa = -1 0 = 1 da/dn mm/cycle da/dn mm/cycle K, MPa m K, MPa m MultiaxialFatigue Darrell Socie, All Rights Reserved 76 of 85
78 Mode I and Mode III Growth 10-3 da/dn, mm/cycle K I K III K I, K III MPa m MultiaxialFatigue Darrell Socie, All Rights Reserved 77 of 85
79 Mode I and Mode II Growth T6 Aluminum Mode I R = 0 Mode II R = -1 da/dn, mm/cycle SNCM Steel Mode I R = -1 Mode II R = K I, K II MPa m MultiaxialFatigue Darrell Socie, All Rights Reserved 78 of 85
80 Fracture Surfaces Bending Torsion MultiaxialFatigue Darrell Socie, All Rights Reserved 79 of 85
81 Mode III Growth Crack growth rate, mm/cycle K = 14.9 K = 12.0 K = 11.0 K = 10.0 K = Crack length, mm MultiaxialFatigue Darrell Socie, All Rights Reserved 80 of 85
82 Fracture Mechanics Models K da dn C K m eq K 8K 8K ( eq I II III ) K K eq eq K K (1 ) K 0. 5 I II 2 2 K K K K 0. 5 I I II III II K eq ( ) (F K eq II E ) 2(1 ) 2 FG 1 k (FE) I n,max ys a a MultiaxialFatigue Darrell Socie, All Rights Reserved 81 of 85
83 Growth of Inclined Cracks Steel 20 tensile growth K II K no growth K shear growth Cracks grow in either tension or shear K I From: Otsuka et.al. Engineering Fracture Mechanics, Vol 7, 1975 MultiaxialFatigue Darrell Socie, All Rights Reserved 82 of 85
84 Otsuka Tensile growth: K cos 2 K I cos K II sin Shear growth: 1 K cos KI sin KII 3cos MultiaxialFatigue Darrell Socie, All Rights Reserved 83 of 85
85 Strain Energy Density Strain energy density at the crack tip: S a K I 2a12KIK II a22kii a33kiii Necessary and sufficient conditions for crack growth: S 0 at o 2 S 0 at 2 o Cyclic strain energy density: [ S a ( ) K K a ( )( K K K K ) 2 o I mean I o II mean I I mean II a o KII mean mean 22 ( ) KII a33( o ) KIII KIII ] Sih, G.C and Barthelemy, B.M. Mixed Mode Fatigue Crack Growth Predictions Engineering Fracture Mechanics, Vol. 13, 1980 MultiaxialFatigue Darrell Socie, All Rights Reserved 84 of 85
86 Summary Many models but no experimental verification for out-of-phase spectrum loads MultiaxialFatigue Darrell Socie, All Rights Reserved 85 of 85
87 Multiaxial Fatigue
NOTCHES AND THEIR EFFECTS. Ali Fatemi - University of Toledo All Rights Reserved Chapter 7 Notches and Their Effects 1
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