Fatigue and Fracture

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4 Proportional Biaxial Z Y principal stresses vary proportionally but do not rotate σ 1 = ασ 2 = βσ 3 X Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 134

5 Nonproportional Multiaxial Z Principal stresses may vary nonproportionally and/or change direction Y X Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 134

Crankshaft 2500 0 45 90 Microstrain 0-2500 Time Multiaxial Fatigue 2003-2013

9 3D stresses 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 8 of 134

Book Multiaxial Fatigue 2003-2013 Darrell Socie, University

11 Outline State of Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 134

13 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 12 of 134

15 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 14 of 134

16 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 15 of 134

19 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 18 of 134

20 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 19 of 134

21 Maximum and Octahedral Shear 18% Maximum shear Octahedral shear 6% σ 3 /σ 1 torsion tension biaxial tension Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 20 of 134

22 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 21 of 134

23 Outline State of Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 22 of 134

24 The Fatigue Process Crack nucleation Small crack growth in an elastic-plastic stress field Macroscopic crack growth in a nominally elastic stress field Final fracture Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 23 of 134

25 Slip Bands Ma, B-T and Laird C. Overview of fatigue behavior in copper sinle crystals II Population, size, distribution and growth Kinetics of stage I cracks for tests at constant strain amplitude, Acta Metallurgica, Vol 37, 1989, Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 24 of 134

Mode I Growth crack growth direction 5 µm Multiaxial Fatigue 2003-2013 Darrell

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

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

10 6 10 7 Fatigue Life, 2N f Multiaxial Fatigue 2003-2013 Darrell

28 Steel - Tension Damage Fraction N/N f Tension Shear Nucleation 100 µm crack Fatigue Life, 2N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 27 of 134

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

29 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 28 of 134

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

30 304 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 29 of 134

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

34 Outline State of Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 134

35 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 34 of 134

37 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 36 of 134

38 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 37 of 134

39 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 38 of 134

40 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 39 of

41 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 40 of 134

42 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 41 of 134

43 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 42 of 134

44 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 43 of 134

46 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 45 of 134

48 Isotropic Hardening σ ε stabilized stress range Failure occurs when the stress range is not elastic Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 47 of 134

49 Multiaxial Kinematic and Isotropic a) b) σ 3 σ 3 Yield domain expands and translates C o O o O o σ 1 R o σ 2 σ 2 σ 1 c) d) σ 3 Loading path σ 3 C L R L O L O o O o ρ σ 1 σ 2 σ 1 σ 2 ρ* stabilized residual stress Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 48 of 134

51 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 50 of 134

52 Model Comparison R = Relative Fatigue Life Goodman Findley Sines Torsion Tension Biaxial Tension Stress ratio, λ Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 51 of 134

53 Outline State of Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 52 of 134

54 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 53 of 134

55 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 54 of 134

56 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 55 of 134

60 Brown and Miller ( continued ) 1 α α α max ( ) γ = γ + 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 59 of 134

67 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 66 of 134

68 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 67 of 134

69 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 68 of 134

70 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 69 of 134

71 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 70 of 134

72 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 71 of 134

73 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 72 of 134

75 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 74 of 134

76 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 75 of 134

77 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 76 of 134

79 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 78 of 134

80 Outline State of Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 79 of 134

81 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 80 of 134

82 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 81 of 134

84 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 83 of 134

86 Findley Model Results τ/2 MPa σ n,max MPa 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 85 of 134

87 Nonproportional Hardening ε x t ε x = ε o sin(ωt) γ xy t In-phase γ xy = (1+ν)ε o sin(ωt) 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 86 of 134

93 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 92 of 134

94 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 93 of 134

102 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 101 of 134

An Example Analysis model Single event 16 input channels 2240 elements From Khosrovaneh, Pattu and Schnaidt Discussion of Fatigue Analysis Techniques for

103 An Example Analysis model Single event 16 input channels 2240 elements From Khosrovaneh, Pattu and Schnaidt Discussion of Fatigue Analysis Techniques for Automotive Applications Presented at SAE Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 102 of 134

104 Biaxial and Uniaxial Solution Uniaxial solution Signed principal stress Critical plane solution Damage Element rank based on critical plane damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 103 of 134

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 134

106 Outline State of Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 105 of 134

107 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 106 of 134

109 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 108 of 134

111 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 110 of 134

Torsion Experiments Multiaxial Fatigue 2003-2013 Darrell Socie,

114 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 113 of 134

115 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 114 of 134

116 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 115 of 134

121 Maximum Tensile Stress Location σ σ τ = σ τ τ = σ τ σ K t = 3 σ 1 = σ σ K t = 3.41 σ 1 = 1.72σ K t = 4 σ 1 = τ Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 120 of 134

122 In and Out of Phase Loading In-phase Out-of-phase K t = 3 K t = 4 Damage location changes with load phasing Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 121 of 134

123 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 122 of 134

124 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 123 of 134

125 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 124 of 134

126 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 125 of 134

Fracture Surfaces in Torsion Circumferencial

128 Neuber s Rule K t S Actual stress Stress (MPa) σ ε K t S K t e = σε Stress calculated with elastic assumptions e S e e = σε For cyclic loading K t e Strain e S 2 = E σ ε Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 127 of 134

129 Multiaxial Neuber s Rule Define Neuber s rule in equivalent variables Stress strain curve ε = Constitutive equation e S 2 σ + E = E σ ε σ K' 1 n' σ σ τ x y xy = f(e,k,n ) ε ε γ x y xy Five equations and six unknowns Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 128 of 134

130 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 129 of 134 Ignore Plasticity Theory 1 1 e 2 e 2 e e ε = ε 1 1 e 3 e 3 e e ε = ε 1 1 e 2 e 2 S S σ = σ 1 1 e 3 e 3 S S σ = σ

132 Glinka Strain energy density e S σ σ ij σ ε ij ij ε ij = e S e e ij ij e ij eij S e Stress (MPa) e e ε Strain Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 131 of 134

133 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 132 of 134 Koettgen-Barkey-Socie γ ε ε = xy y x xy e y e x e T S S f o (E o,k o,n o ) E K n Material Yield Surface Structural Yield Surface K o n o E o γ ε ε = τ σ σ xy y x xy y x f(e,k,n ) σ ε ε S e

134 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 133 of 134

136 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 135 of 134

137 Multiaxial Fatigue

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

Multiaxial Fatigue. Professor Darrell F. Socie. Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Multiaxial Fatigue Professor Darrell F. Socie Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign 2001-2011 Darrell Socie, All Rights Reserved Contact Information