Multiaxial Fatigue. Professor Darrell F. Socie. Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign

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2 Contact Information Darrell Socie Mechanical Science and Engineering 1206 West Green Urbana, Illinois USA Tel: Fax: Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 125

3 Outline State of Stress ( Chapter 1 ) Fatigue Mechanisms ( Chapter 3 ) Stress Based Models ( Chapter 5 ) Strain Based Models ( Chapter 6 ) Fracture Mechanics Models ( Chapter 7 ) Nonproportional Loading ( Chapter 8 ) Notches ( Chapter 9 ) Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 2 of 125

5 Stress Components Z σ z τ zy τ zx τ xz τ yz τ xy τ yx σ y σ x Y X Six stresses and six strains Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 125

7 Principal Stresses τ 13 σ 3 τ12 τ 23 σ 2 σ 1 σ 3 - σ 2 ( σ X + σ Y + σ Z ) + σ(σ X σ Y + σ Y σ Z σ X σ Z -τ 2 XY - τ2 YZ -τ2 XZ ) - (σ X σ Y σ Z + 2τ XY τ YZ τ XZ - σ X τ 2 YZ - σ Y τ2 ZX - σ Z τ2 XY ) = 0 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 125

8 Stress and Strain Distributions 100 % of applied stress θ Stresses are nearly the same over a 10 range of angles Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 125

11 Biaxial Tension σ 3 τ γ / 2 σ 3 σ ε σ 1 = σ x σ 1 = σ 2 ε 1 = ε 2 σ 2 = σ y ε 3 2ν = 1 ν ε Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 125

12 Shear Stresses σ 3 σ 3 σ 2 σ 1 Maximum shear stress σ 2 σ 1 Octahedral shear stress σ σ Mises: σ = 3 3 τ oct τoct = τ13 = τ τ 13= τ ( ) ( ) ( ) 2 oct= σ1 σ3 + σ1 σ 2 + σ2 σ3 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 11 of 125

13 State of Stress Summary Stresses acting on a plane Principal stress Maximum shear stress Octahedral shear stress Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 12 of 125

Slip Bands Multiaxial Fatigue 2001-2011 Darrell Socie,

18 Mode I, Mode II, and Mode III Mode I opening Mode II in-plane shear Mode III out-of-plane shear Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 17 of 125

Mode I Growth crack growth direction 5 µm Multiaxial Fatigue 2001-2011 Darrell

Mode II Growth shear stress slip bands 10 µm crack growth direction Multiaxial Fatigue

304 Stainless Steel - Torsion 1.0 Damage Fraction N/N f 0.8 0.6 0.4 0.

23 304 Stainless Steel - Torsion 1.0 Damage Fraction N/N f Shear Tension 0 Nucleation Fatigue Life, 2N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 22 of 125

Inconel 718 - Tension Damage Fraction N/N f 1.0 0.8 0.6 0.4 0.

1045 Steel - Torsion 1.0 Tension Damage Fraction N/N f f 0.8 0.

2 Shear 0 Nucleation 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue

27 1045 Steel - Torsion 1.0 Tension Damage Fraction N/N f f Shear 0 Nucleation Fatigue Life, 2N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 26 of 125

1.0 1045 Steel - Tension Damage Fraction N/N f 0.8 0.6 0.4 0.

Fatigue Life, 2N f Multiaxial Fatigue 2001-2011 Darrell Socie,

29 Fatigue Mechanisms Summary Fatigue cracks nucleate in shear Fatigue cracks grow in either shear or tension depending on material and state of stress Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 28 of 125

31 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 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 125

32 Test Results Cyclic tension with static tension Cyclic torsion with static torsion Cyclic tension with static torsion Cyclic torsion with static tension Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 125

33 Cyclic Tension with Static Tension 1.5 Axial stress Fatigue strength Mean stress Yield strength 1.5 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 32 of 125

34 Cyclic Torsion with Static Torsion 1.5 Shear Stress Amplitude Shear Fatigue Strength Maximum Shear Stress Shear Yield Strength Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of

35 Cyclic Tension with Static Torsion 1.5 Bending Stress Bending Fatigue Strength Static Torsion Stress Torsion Yield Strength 3.0 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 125

36 Cyclic Torsion with Static Tension 1.5 Torsion shear stress Shear fatigue strength Axial mean stress 1.5 Yield strength Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 125

37 Conclusions Tension mean stress affects both tension and torsion Torsion mean stress does not affect tension or torsion Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 125

38 Sines τ 2 oct + α(3σ h ) =β 1 6 ( σ x α( σ σ mean x y ) 2 + σ + ( σ mean y x + σ σ z mean z ) 2 ) = β + ( σ y σ z ) 2 + 6( τ 2 xy + τ 2 xz + τ 2 yz ) + Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 125

40 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 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 125

43 Stress Based Models Summary Sines: Findley: τ 2 oct τ α(3σ h ) =β kσ n = max f Dang Van: τ σ ( t) + a ( t) = b h Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 42 of 125

44 Strain Based Models Plastic Work Brown and Miller Fatemi and Socie Smith Watson and Topper Liu Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 43 of 125

45 Octahedral Shear Strain 0.1 Plastic octahedral shear strain range 0.01 Tension Torsion Cycles to failure Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 125

46 Plastic Work Plastic Work per Cycle, MJ/m A T T A A T T A T Torsion A Axial T A T A T Fatigue Life, N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 125

50 Brown and Miller ( continued ) 1 α α α ( ) γ = γ max + S ε n γ max 2 + S ε = A n σ ' f 2σ E n, mean b ' f + ε f ( 2N ) B ( 2N ) f c Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 125

57 Liu Virtual strain energy for both mode I and mode II cracking W I = ( σ n ε n ) max + ( τ γ) W ' 2 ' ' b+ c 4σf 2b I = 4σfεf(2N f ) + (2Nf ) E W II = ( σ n ε n ) + ( τ γ) max W ' 2 ' ' bo+ co 4τf 2bo II = 4τfγ f(2n f ) + (2Nf ) G Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 125

58 Cyclic Torsion Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Shear Damage Tensile Damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 57 of 125

59 Cyclic Torsion with Static Tension Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Tension Shear Damage Tensile Damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 58 of 125

60 Cyclic Torsion with Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Shear Damage Tensile Damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 125

61 Cyclic Torsion with Tension and Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Hoop Tension Shear Damage Tensile Damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 60 of 125

62 Test Results Load Case γ/2 σ hoop MPa σ axial MPa N f Torsion ,200 with tension ,300 with compression ,000 with tension and compression ,200 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 61 of 125

63 Conclusions All critical plane models correctly predict these results Hydrostatic stress models can not predict these results Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 62 of 125

65 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 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 64 of 125

66 Strain Based Models Summary Two separate models are needed, one for tensile growth and one for shear growth Cyclic plasticity governs stress and strain ranges Mean stress effects are a result of crack closure on the critical plane Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 125

67 Separate Tensile and Shear Models τ σ 3 σ 1 σ σ 2 σ 1 = τ xy Inconel 1045 steel stainless steel Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 125

69 γ 2 Mean Stresses ' σ f σ mean ε eq = W = I E f b ' f ( 2N ) + ε ( 2N ) ' max f n b ' c + S ε ) n= ( S) (2N f ) + ( S) εf (2N f γ 2 σ σ 1+ k σ n,max y [( σ ε ) + ( τ γ) ] n σ n 2σ E ' τf = (2N f ) G max bo + γ ' f f (2N ) ' 2 ε1 σf 2b ' ' b+ c n = (2N f ) + σfεf (2Nf ) 2 E f 2 1 R c co Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 125

71 Mode I, Mode II, and Mode III Mode I opening Mode II in-plane shear Mode III out-of-plane shear Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 70 of 125

73 Biaxial Mode I Growth 10-3 σ = 193 MPa λ = -1 0 λ = σ = 386 MPa λ = -1 0 λ = 1 da/dn mm/cycle da/dn mm/cycle K, MPa m K, MPa m Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 72 of 125

76 Fracture Mechanism Map 60 No Cracks Yield Strength, MPa Spiral Cracks Longitudinal Transverse Cracks Hardness Rc Shear Stress Amplitude, MPa Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 125

78 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 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 77 of 125

79 Fracture Mechanics Models K da dn = C ( ) m K eq [ ] K + 8 K + 8 K (1 ν eq = I II III ) K K eq eq [ ] K + K + (1+ν ) K 0. 5 = I II [ ] 2 2 K + K K + K 0. 5 = I I II III II K eq ( ε) = (F K eq II E γ) 2(1 + ν) 2 + (FE ε) σ I 2 n,max ( ε) = FG γ 1+ k πa σ ys 0.5 πa Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 78 of 125

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 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 80 of 125

82 Fracture Mechanics Models Summary Multiaxial loading has little effect in Mode I Crack closure makes Mode II and Mode III calculations difficult Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 81 of 125

83 Nonproportional Loading In and Out-of-phase loading Nonproportional cyclic hardening Variable amplitude Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 82 of 125

84 In and Out-of-Phase Loading γ ε x ε x t ε x = ε o sin(ωt) 1 + ν ε γ xy ε y γ xy ε x γ xy γ xy = (1+ν)ε o sin(ωt) t In-phase ε x = ε o cos(ωt) t γ xy = (1+ν)ε o sin(ωt) t γ γ 1 + ν ε Out-of-phase γ Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 83 of 125

86 Loading Histories out-of-phase γ xy /2 γ xy /2 diamond ε x ε x square γ xy /2 cross γ xy /2 ε x ε x Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 85 of 125

88 Findley Model Results τ/2 MPa σ MPa n,max τ/ σ n,max N/N ip in - phase out - of - phase diamond square cross - tension cycle cross - torsion cycle out-of-phase γ xy /2 diamond γ xy /2 square γ xy /2 cross γ xy /2 ε x ε x ε x ε x in-phase Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 87 of 125

89 Nonproportional Hardening ε x t ε x = ε o sin(ωt) γ xy t γ xy = (1+ν)ε o sin(ωt) In-phase ε x γ xy t t ε x = ε o cos(ωt) γ xy = (1+ν)ε o sin(ωt) Out-of-phase Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 88 of 125

95 Stress-Strain Response (continued) Case 7 Case 9 Case Shear Stress ( MPa ) Case 11 Case 12 Case Axial Stress ( MPa ) Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 125

96 Maximum Stress 2000 All tests have the same strain ranges Equivalent Stress, MPa Fatigue Life, N f Nonproportional hardening results in lower fatigue lives Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 95 of 125

104 Fatigue Calculations Load or strain history Cyclic plasticity model Stress and strain tensor Search for critical plane Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 103 of 125

105 Nonproportional Loading Summary Nonproportional cyclic hardening increases stress levels Critical plane models are used to assess fatigue damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 104 of 125

106 Notches Stress and strain concentrations Nonproportional loading and stressing Fatigue notch factors Cracks at notches Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 105 of 125

108 Stress Concentration Factors 6 Stress Concentration Factor Torsion Bending d ρ D D/d Notch Root Radius, ρ/d Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 107 of 125

110 Stresses at the Hole λ = 1 σ θ σ Angle λ = -1 λ = 0 Stress concentration factor depends on type of loading Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 109 of 125

Torsion Experiments Multiaxial Fatigue 2001-2011 Darrell Socie,

113 Multiaxial Loading Uniaxial loading that produces multiaxial stresses at notches Multiaxial loading that produces uniaxial stresses at notches Multiaxial loading that produces multiaxial stresses at notches Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 112 of 125

114 Thickness Effects Longitudinal Tensile Strain Transverse Compression Strain Thickness 50 mm 30 mm 15 mm 7 mm y z x 100 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 113 of 125

118 Torsion Loading M X t 1 σ T σ 1 t 3 M T σ z σ 1 = σ z M X σ T σ T σ 1 = σ T t 2 t 4 M T σ z t 1 t 2 t 3 t 4 σ T σ 1 σ T Out-of-phase shear loading is needed to produce nonproportional stressing Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 117 of 125

120 Fatigue Notch Factors 6 Stress concentration factor K f Torsion d ρ K f Bending K t Bending K t Torsion ρ Notch root radius, d D D d = 2.2 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 119 of 125

121 Fatigue Notch Factors ( continued ) Calculated K f bending torsion conservative non-conservative Peterson s Equation K f K T 1 = 1 + a 1 + r Experimental K f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 120 of 125

Fracture Surfaces in Torsion Circumferencial

123 Stress Intensity Factors λ = 1 λ = F 0.75 λ = 1 σ R 0.50 λσ λσ a 0.25 σ a R da dn ( ) m = C = F σ πa K eq K I Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 122 of 125

125 Notches Summary Uniaxial loading can produce multiaxial stresses at notches Multiaxial loading can produce uniaxial stresses at notches Multiaxial stresses are not very important in thin plate and shell structures Multiaxial stresses are not very important in crack growth Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 124 of 125

126 Final Summary Fatigue is a planar process involving the growth of cracks on many size scales Critical plane models provide reasonable estimates of fatigue damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 125 of 125

127 Multiaxial Fatigue University of Illinois at Urbana-Champaign

Fatigue and Fracture

Fatigue and Fracture Multiaxial Fatigue Professor Darrell F. Socie Mechanical Science and Engineering University of Illinois 2004-2013 Darrell Socie, All Rights Reserved When is Multiaxial Fatigue Important?