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

Contact Information Darrell Socie Mechanical Science and Engineering 1206 West Green Urbana, Illinois 61801 USA d-socie@uiuc.edu Tel: 217 333 7630 Fax: 217 333 5634 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 2 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 125

Stress and Strain Distributions 100 % of applied stress 90 80-20 -10 0 10 20 θ Stresses are nearly the same over a 10 range of angles Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 125

Shear Stresses σ 3 σ 3 σ 2 σ 1 Maximum shear stress σ 2 σ 1 Octahedral shear stress σ σ 1 2 3 Mises: σ = 3 3 τ oct τoct = τ13 = 0. 94 τ13 2 2 2 2 2 1 3 τ 13= τ ( ) ( ) ( ) 2 oct= σ1 σ3 + σ1 σ 2 + σ2 σ3 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 11 of 125

State of Stress Summary Stresses acting on a plane Principal stress Maximum shear stress Octahedral shear stress Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 12 of 125

Fatigue Mechanisms Crack nucleation Fracture modes Crack growth State of stress effects Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 13 of 125

Mode I, Mode II, and Mode III Mode I opening Mode II in-plane shear Mode III out-of-plane shear Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 17 of 125

304 Stainless Steel - Torsion 1.0 Damage Fraction N/N f 0.8 0.6 0.4 0.2 Shear Tension 0 Nucleation 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, 2N f Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 22 of 125

1.0 304 Stainless Steel - Tension Damage Fraction N/N f 0.8 0.6 0.4 0.2 Tension Nucleation 0 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, 2N f Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 23 of 125

Inconel 718 - Torsion 1.0 Tension Damage Fraction N/N f 0.8 0.6 0.4 0.2 0 Shear Nucleation 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, 2N f Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 24 of 125

Inconel 718 - Tension Damage Fraction N/N f 1.0 0.8 0.6 0.4 0.2 0 Tension Shear Nucleation 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, 2N f Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 25 of 125

1045 Steel - Torsion 1.0 Tension Damage Fraction N/N f f 0.8 0.6 0.4 0.2 Shear 0 Nucleation 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, 2N f Multiaxial Fatigue 2001-2011 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 0.2 Shear Tension Nucleation 0 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, 2N f Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 27 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 28 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 125

Test Results Cyclic tension with static tension Cyclic torsion with static torsion Cyclic tension with static torsion Cyclic torsion with static tension Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 32 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 125 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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 125

Conclusions Tension mean stress affects both tension and torsion Torsion mean stress does not affect tension or torsion Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 125

Dang Van ( continued ) τ τ(t) + aσ h (t) = b τ Failure predicted σ h σ h Loading path Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 41 of 125

Stress Based Models Summary Sines: Findley: τ 2 oct τ 2 + + α(3σ h ) =β kσ n = max f Dang Van: τ σ ( t) + a ( t) = b h Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 42 of 125

Strain Based Models Plastic Work Brown and Miller Fatemi and Socie Smith Watson and Topper Liu Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 43 of 125

Octahedral Shear Strain 0.1 Plastic octahedral shear strain range 0.01 Tension Torsion 0.001 10 10 2 10 3 10 4 10 5 Cycles to failure Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 125

Plastic Work Plastic Work per Cycle, MJ/m 3 100 10 A T T A A T T A T Torsion A Axial 0 90 180 135 45 30 1 10 2 10 3 10 4 T A T A T Fatigue Life, N f Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 125

Brown and Miller Fatigue Life, Cycles 2 x10 3 10 3 5 x10 2 2 x10 2 10 2 0.0 γ = 0.03 0.005 0.01 Normal Strain Amplitude, ε n Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 46 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 52 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 125

Cyclic Torsion Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Shear Damage Tensile Damage Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 57 of 125

Cyclic Torsion with Static Tension Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Tension Shear Damage Tensile Damage Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 58 of 125

Cyclic Torsion with Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Shear Damage Tensile Damage Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 125

Cyclic Torsion with Tension and Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Hoop Tension Shear Damage Tensile Damage Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 60 of 125

Test Results Load Case γ/2 σ hoop MPa σ 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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 61 of 125

Conclusions All critical plane models correctly predict these results Hydrostatic stress models can not predict these results Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 62 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 64 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 125

Separate Tensile and Shear Models τ σ 3 σ 1 σ σ 2 σ 1 = τ xy Inconel 1045 steel stainless steel Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 125

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= (1.3 + 0.7S) (2N f ) + (1.5 + 0.5S) 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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 125

Biaxial Mode I Growth 10-3 σ = 193 MPa λ = -1 0 λ = 1 10-3 σ = 386 MPa λ = -1 0 λ = 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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 72 of 125

Fracture Mechanism Map 60 No Cracks Yield Strength, MPa 1500 1000 Spiral Cracks Longitudinal Transverse Cracks 50 40 30 Hardness Rc 200 300 400 500 Shear Stress Amplitude, MPa Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 77 of 125

Fracture Mechanics Models K da dn = C ( ) m K eq [ ] 4 4 4 K + 8 K + 8 K (1 ν 0. 25 eq = I II III ) K K eq eq [ ] 2 2 2 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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 78 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 80 of 125

Fracture Mechanics Models Summary Multiaxial loading has little effect in Mode I Crack closure makes Mode II and Mode III calculations difficult Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 81 of 125

Nonproportional Loading In and Out-of-phase loading Nonproportional cyclic hardening Variable amplitude Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 82 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 83 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 85 of 125

Findley Model Results τ/2 MPa σ MPa n,max τ/2 + 0.3 σ n,max N/N ip in - phase 353 250 428 1.0 90 out - of - phase 250 500 400 2.0 diamond 250 500 400 2.0 square 353 603 534 0.11 cross - tension cycle 250 250 325 16 cross - torsion cycle 250 0 250 216 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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 87 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 88 of 125

Critical Plane Proportional N f = 38,500 N f = 310,000 600 Out-of-phase N f = 3,500 N f = 40,000 600-0.004 0.004-0.004 0.004-600 -600 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 91 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 93 of 125

Stress-Strain Response (continued) 300 300 300 Case 7 Case 9 Case 10 150 150 150 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 11 Case 12 Case 13 150 150 150 0 0 0-150 -150-150 -300-300 -300-600 -300 0 300 600-600 -300 0 300 600-600 -300 0 300 600 Axial Stress ( MPa ) Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 125

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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 95 of 125

Nonproportional Example Case A Case B Case C Case D σ x σ x σ x σ x σ y σ y σ y σ y Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 96 of 125

Stress-Strain on 30 and 60 Planes 30º plane 300 60º plane 300-0.003 0.003 ε 30-0.003 0.003 ε 60-300 -300 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 100 of 125

Stress-Strain on 120 and 150 Planes 120º plane 300 150º plane 300-0.003 0.003-0.003 0.003 ε 120 ε 150-300 -300 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 101 of 125

Fatigue Calculations Load or strain history Cyclic plasticity model Stress and strain tensor Search for critical plane Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 103 of 125

Nonproportional Loading Summary Nonproportional cyclic hardening increases stress levels Critical plane models are used to assess fatigue damage Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 104 of 125

Notches Stress and strain concentrations Nonproportional loading and stressing Fatigue notch factors Cracks at notches Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 105 of 125

Stress Concentration Factors 6 Stress Concentration Factor 5 4 3 2 1 Torsion Bending d ρ D D/d 2.20 1.20 1.04 0 0.025 0.050 0.075 0.100 0.125 Notch Root Radius, ρ/d Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 107 of 125

Stresses at the Hole 4 3 2 λ = 1 σ θ σ 1 0-1 -2-3 -4 30 60 90 120 150 180 Angle λ = -1 λ = 0 Stress concentration factor depends on type of loading Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 109 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 112 of 125

Thickness Effects 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 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 113 of 125

Bending Moments M A B C D 2.82 1 1 2.00 3 2 1.41 2 1 1.00 2 0.71 2 M 5 = M 5 A B C D M 2.49 2.85 2.31 2.84 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 116 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 117 of 125

Fatigue Notch Factors 6 Stress concentration factor 5 4 3 2 1 0 K f Torsion d ρ K f Bending K t Bending K t Torsion 0.025 0.050 0.075 ρ 0.100 0.125 Notch root radius, d D D d = 2.2 Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 119 of 125

Fatigue Notch Factors ( continued ) Calculated K f 2.5 2.0 1.5 bending torsion conservative non-conservative Peterson s Equation K f K T 1 = 1 + a 1 + r 1.0 1.0 1.5 2.0 2.5 Experimental K f Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 120 of 125

Stress Intensity Factors 1.50 1.25 λ = 1 λ = 0 1.00 F 0.75 λ = 1 σ R 0.50 λσ λσ a 0.25 σ 0.00 1.00 1.50 2.00 a R 2.50 3.00 da dn ( ) m = C = F σ πa K eq K I Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 122 of 125

Crack Growth From a Hole 100 λ = -1 λ = 0 λ = 1 Crack Length, mm 80 60 40 20 0 0 2 x 10 6 4 x 10 6 6 x 10 6 8 x 10 6 Cycles Multiaxial Fatigue 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 123 of 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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 124 of 125

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 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 125 of 125

Multiaxial Fatigue University of Illinois at Urbana-Champaign