Multiaxial Fatigue Professor Darrell Socie 2008-2014 Darrell Socie, All Rights Reserved
Outline Stresses around holes Crack Nucleation Crack Growth MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 1 of 85
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 2008-2014 Darrell Socie, All Rights Reserved 2 of 85
3D stresses Longitudinal Tensile Strain 0 0.004 0.008 0.012 Transverse Compression Strain 0.001 0.002 0.003 0.004 0.005 Thickness 50 mm 30 mm 15 mm 7 mm y z x 100 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 3 of 85
Notch Stresses y z x t x z x z 7 0.01-0.005 63.5 0 15 0.01-0.003 70.6 14.1 30 0.01-0.002 73.0 21.8 50 0.01-0.001 75.1 29.3 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 4 of 85
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 2008-2014 Darrell Socie, All Rights Reserved 5 of 85
Hole in a Plate a r r r r r MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 6 of 85
Stresses at the Hole 4 3 2 = 1 1 0-1 -2-3 -4 30 60 90 120 150 180 Angle = -1 = 0 Stress concentration factor depends on type of loading MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 7 of 85
Combined Loading = MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 8 of 85
Maximum Tensile Stress Location 32 45 K t = 3 1 = K t = 3.41 1 = 1.72 K t = 4 1 = MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 9 of 85
In and Out of Phase Loading In-phase Out-of-phase K t = 3 K t = 4 Damage location changes with load phasing MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 10 of 85
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 2008-2014 Darrell Socie, All Rights Reserved 11 of 85
Notched Shaft Loading M T M X P M Y z z MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 12 of 85
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 2008-2014 Darrell Socie, All Rights Reserved 13 of 85
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 2008-2014 Darrell Socie, All Rights Reserved 14 of 85
Shear Stresses Around Hole = 0 1.5 = 0 r xy 1.0 0.5 0 1 2 r 3 4 5 a MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 15 of 85
Torsion Experiments MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 16 of 85
Shear Stresses Around Hole r = 1.33 = 0 r xy 1.4 1.0 0.6 r 1.33 a 0.2 0 20 40 60 80 100 120 140 160 180 Angle MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 17 of 85
Stress Intensity Factors F 1.50 1.25 1.00 0.75 R 0.50 a 0.25 0.00 1.00 1.50 2.00 a 2.50 3.00 R da dn C K m eq K I F a MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 18 of 85
Cracks from Holes Crack nucleates in shear Mixed mode growth? Tensile mode growth? MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 19 of 85
Summary Cracks nucleate in a uniaxial stress field and then grow in a mixed tensile/shear stress field MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 20 of 85
Outline Stresses around holes Crack Nucleation Stress Based Models Strain Based Models Crack Growth MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 21 of 85
Fatigue Mechanisms Summary Fatigue cracks nucleate in shear Fatigue cracks grow in either shear or tension depending on material and state of stress MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 22 of 85
Stress Based Models Sines Findley Dang Van MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 23 of 85
Bending Torsion Correlation Shear stress in bending 1/2 Bending fatigue limit 1.0 0.5 0 Shear stress Octahedral stress Principal stress 0 0.5 1.0 1.5 2.0 Shear stress in torsion 1/2 Bending fatigue limit MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 24 of 85
Test Results Cyclic tension with static tension Cyclic torsion with static torsion Cyclic tension with static torsion Cyclic torsion with static tension MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 25 of 85
Cyclic Tension with Static Tension 1.5 Axial stress Fatigue strength 1.0 0.5-1.5-1.0-0.5 0 0.5 1.0 Mean stress Yield strength 1.5 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 26 of 85
Cyclic Torsion with Static Torsion 1.5 Shear Stress Amplitude Shear Fatigue Strength 1.0 0.5 0 0.5 1.0 Maximum Shear Stress Shear Yield Strength MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 27 of 85 1.5
Cyclic Tension with Static Torsion 1.5 Bending Stress Bending Fatigue Strength 1.0 0.5 0 2.0 1.0 Static Torsion Stress Torsion Yield Strength 3.0 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 28 of 85
Cyclic Torsion with Static Tension 1.5 Torsion shear stress Shear fatigue strength 1.0 0.5-1.5-1.0-0.5 0 0.5 1.0 Axial mean stress 1.5 Yield strength MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 29 of 85
Conclusions Tension mean stress affects both tension and torsion Torsion mean stress does not affect tension or torsion MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 30 of 85
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 2008-2014 Darrell Socie, All Rights Reserved 31 of 85
Findley 2 k n max f tension torsion MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 32 of 85
Bending Torsion Correlation Shear stress in bending 1/2 Bending fatigue limit 1.0 0.5 0 Shear stress Octahedral stress Principal stress 0 0.5 1.0 1.5 2.0 Shear stress in torsion 1/2 Bending fatigue limit MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 33 of 85
Dang Van ij (M,t) E ij (M,t) () t a () t b h m ij (m,t) ij (m,t) V(M) MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 34 of 85
Isotropic Hardening stabilized stress range Failure occurs when the stress range is not elastic MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 35 of 85
Multiaxial Kinematic and Isotropic a) b) 3 3 Yield domain expands and translates C o O o O o 1 R o 2 2 1 c) d) Loading path 3 3 C L R L O L O o O o 1 2 2 1 * stabilized residual stress MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 36 of 85
Dang Van ( continued ) ta h t= b Failure predicted h h Loading path MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 37 of 85
Stress Based Models Summary Sines: Findley: 2 oct 2 (3 h ) k n max f Dang Van: () t a () t b h MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 38 of 85
Model Comparison R = -1 10 Relative Fatigue Life 1 0.1 0.01 Goodman Findley Sines 0.001-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 Torsion Tension Biaxial Tension Stress ratio, MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 39 of 85
Outline Stresses around holes Crack Nucleation Stress Based Models Strain Based Models Crack Growth MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 40 of 85
Strain Based Models Brown and Miller Fatemi and Socie Smith Watson and Topper MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 41 of 85
Brown and Miller Fatigue Life, Cycles 2 x10 3 10 3 5 x10 2 2 x10 2 10 2 0.0 = 0.03 0.005 0.01 Normal Strain Amplitude, n MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 42 of 85
Case A and B Growth along the surface Growth into the surface MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 43 of 85
Brown and Miller ( continued ) Uniaxial Equibiaxial MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 44 of 85
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 2008-2014 Darrell Socie, All Rights Reserved 45 of 85
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 2008-2014 Darrell Socie, All Rights Reserved 46 of 85
/ 3 Loading Histories C F G H I J MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 47 of 85
Crack Directions MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 48 of 85
Hypothesis Fatigue damage is planar in nature The material finds a critical plane for microcrack growth. % of applied stress 100 90 80-20 -10 0 10 20 Stresses are nearly the same over a 10 range of angles MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 49 of 85
Stresses on the Planes Shear Tension MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 50 of 85
Fatemi and Socie MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 51 of 85
Fatigue Lives 12,899 12,121 3 9,500 5,963 3 14,117 16,062 3 C F G 3 3 3 7,676 8,020 7,419 9,139 H I J 5,149 3,488 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 52 of 85
Crack Length Observations 2.5 F-495 H-491 Crack Length, mm 2 1.5 1 J-603 I-471 C-399 G-304 0.5 0 0 2000 4000 6000 8000 10000 12000 14000 Cycles MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 53 of 85
Fatemi and Socie 1 k 2 n,max y ' f (2N f ) G bo ' f (2N ) f co MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 54 of 85
Torsion Tests 3 1 2 1 = xy MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 55 of 85
304 Stainless Steel 1 1 1 1 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 56 of 85
Smith Watson Topper MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 57 of 85
SWT ' 2 1 f 2b ' ' bc n (2N f ) ff(2nf ) 2 E MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 58 of 85
Loading Histories 0 1 2 3 4 5 6 7 8 9 10 11 12 13 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 59 of 85
Stress-Strain Response 300 150 Case 1 300 150 Case 2 300 150 Case 3 Shear Stress ( MPa ) 0 0 0-150 -150-150 -300-300 -300-600 -300 0 300 600-600 -300 0 300 600-600 -300 0 300 600 300 300 300 Case 4 Case 5 Case 6 150 150 150 0 0 0-150 -150-150 -300-300 -300-600 -300 0 300 600-600 -300 0 300 600-600 -300 0 300 600 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 60 of 85
Maximum Stress 2000 All tests have the same strain ranges Equivalent Stress, MPa 1000 200 10 2 10 3 10 4 Fatigue Life, N f Nonproportional hardening results in lower fatigue lives MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 61 of 85
Summary Cracks nucleate in shear and then grow in either shear or tension depending on the material and state of stress MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 62 of 85
Separate Tensile and Shear Models Cyclic shear strains Cyclic tensile strains Normal stresses open and close microcracks MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 63 of 85
Shear Growth shear stress slip bands 10 m crack growth direction ( From Murakami ) MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 64 of 85
Tensile Growth crack growth direction 5 m MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 65 of 85
Cyclic Torsion Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Shear Damage Tensile Damage MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 66 of 85
Cyclic Torsion with Static Tension Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Tension Shear Damage Tensile Damage MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 67 of 85
Cyclic Torsion with Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Shear Damage Tensile Damage MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 68 of 85
Cyclic Torsion with Tension and Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Hoop Tension Shear Damage Tensile Damage MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 69 of 85
Test Results Load Case /2 hoop MPa axial MPa N f Torsion 0.0054 0 0 45,200 with tension 0.0054 0 450 10,300 with compression 0.0054 0-500 50,000 with tension and compression 0.0054 450-500 11,200 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 70 of 85
Conclusions All critical plane models correctly predict these results Hydrostatic stress models can not predict these results MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 71 of 85
Loading History 0.003 Shear strain Axial strain 0.006-0.003 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 72 of 85
Model Comparison Summary of calculated fatigue lives Model Equation Life Epsilon 6.5 14,060 Garud 6.7 5,210 Ellyin 6.17 4,450 Brown-Miller 6.22 3,980 SWT 6.24 9,930 Liu I 6.41 4,280 Liu II 6.42 5,420 Chu 6.37 3,040 Gamma 26,775 Fatemi-Socie 6.23 10,350 Glinka 6.39 33,220 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 73 of 85
Outline Stresses around holes Crack Nucleation Crack Growth MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 74 of 85
Mode I and Mode II Surface Cracks Mode II Mode I MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 75 of 85
Biaxial Mode I Growth 10-3 = 193 MPa = -1 0 = 1 10-3 = 386 MPa = -1 0 = 1 da/dn mm/cycle 10-4 10-5 da/dn mm/cycle 10-4 10-5 10-6 10-6 10 20 50 100 200 10 20 50 100 200 K, MPa m K, MPa m MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 76 of 85
Mode I and Mode III Growth 10-3 da/dn, mm/cycle 10-4 10-5 K I K III 10-6 5 10 20 50 100 K I, K III MPa m MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 77 of 85
Mode I and Mode II Growth 10-2 10-3 7075 T6 Aluminum Mode I R = 0 Mode II R = -1 da/dn, mm/cycle 10-4 10-5 10-6 10-7 SNCM Steel Mode I R = -1 Mode II R = -1 1 10 100 K I, K II MPa m MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 78 of 85
Fracture Surfaces Bending Torsion MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 79 of 85
Mode III Growth Crack growth rate, mm/cycle 10-4 10-5 10-6 10-7 10-8 K = 14.9 K = 12.0 K = 11.0 K = 10.0 K = 8.2 10-9 0.2 0.4 0.6 0.8 1.0 Crack length, mm MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 80 of 85
Fracture Mechanics Models K da dn C K m eq 4 4 4 K 8K 8K (1 0. 25 eq I II III ) K K eq eq 2 2 2 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 2 0.5 a a MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 81 of 85
Growth of Inclined Cracks 25 1010 Steel 20 tensile growth K II 15 10 5 K no growth K shear growth Cracks grow in either tension or shear 0 5 10 15 20 25 K I From: Otsuka et.al. Engineering Fracture Mechanics, Vol 7, 1975 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 82 of 85
Otsuka Tensile growth: K cos 2 K I cos 2 2 3 2 K II sin Shear growth: 1 K cos KI sin KII 3cos 1 2 2 2 MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 83 of 85
Strain Energy Density Strain energy density at the crack tip: S a 2 2 2 11K 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 11 12 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 2008-2014 Darrell Socie, All Rights Reserved 84 of 85
Summary Many models but no experimental verification for out-of-phase spectrum loads MultiaxialFatigue 2008-2014 Darrell Socie, All Rights Reserved 85 of 85
Multiaxial Fatigue