Fatigue and Fracture - PDF Free Download

When is Multiaxial Fatigue Important? Complex state of stress Complex out of phase loading Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 134

Proportional Biaxial Z Y principal stresses vary proportionally but do not rotate σ 1 = ασ 2 = βσ 3 X Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 134

Nonproportional Multiaxial Z Principal stresses may vary nonproportionally and/or change direction Y X Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 134

Shear and Normal Strains 0.005 0º 90º γ 0.005 γ -0.0025 0.0025 ε -0.0025 0.0025 ε -0.005-0.005 Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 134

Shear and Normal Strains 0.005 45º 135º γ 0.005 γ -0.0025 0.0025 ε -0.0025 0.0025 ε -0.005-0.005 Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 134

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 Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 12 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 14 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 15 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 18 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 19 of 134

Maximum and Octahedral Shear 18% 1.0 0.8 0.6 0.4 0.2 Maximum shear Octahedral shear 6% -1-0.5 0 0.5 1 σ 3 /σ 1 torsion tension biaxial tension Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 20 of 134

State of Stress Summary Stresses acting on a plane Principal stress Maximum shear stress Octahedral shear stress Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 21 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 23 of 134

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, 337-348 Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 24 of 134

1.0 1045 Steel - Tension Damage Fraction N/N f 0.8 0.6 0.4 0.2 0 Tension Shear Nucleation 100 µm crack 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, 2N f Multiaxial Fatigue 2003-2013 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 0.6 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 2003-2013 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 0.2 Shear Tension 0 Nucleation 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, 2N f Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 29 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 32 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 134

Test Results Cyclic tension with static tension Cyclic torsion with static torsion Cyclic tension with static torsion Cyclic torsion with static tension Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 134 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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 41 of 134

Conclusions Tension mean stress affects both tension and torsion Torsion mean stress does not affect tension or torsion Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 42 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 43 of 134

Isotropic Hardening σ ε stabilized stress range Failure occurs when the stress range is not elastic Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 47 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 48 of 134

Dang Van ( continued ) τ τ(t) + aσ h (t) = b τ Failure predicted σ h σ h Loading path Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 50 of 134

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, λ Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 51 of 134

Strain Based Models Plastic Work Brown and Miller Fatemi and Socie Smith Watson and Topper Liu Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 53 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 54 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 55 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 62 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 134

Cyclic Torsion Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Shear Damage Tensile Damage Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 67 of 134

Cyclic Torsion with Static Tension Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Tension Shear Damage Tensile Damage Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 134

Cyclic Torsion with Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Shear Damage Tensile Damage Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 69 of 134

Cyclic Torsion with Tension and Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Hoop Tension Shear Damage Tensile Damage Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 70 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 71 of 134

Conclusions All critical plane models correctly predict these results Hydrostatic stress models can not predict these results Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 72 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 74 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 134

Separate Tensile and Shear Models τ σ 3 σ 1 σ σ 2 σ 1 = τ xy Inconel 1045 steel stainless steel Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 78 of 134

Nonproportional Loading In and Out-of-phase loading Nonproportional cyclic hardening Variable amplitude Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 80 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 81 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 83 of 134

Findley Model Results τ/2 MPa σ n,max MPa 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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 85 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 86 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 89 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 91 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 92 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 93 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 98 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 99 of 134

Fatigue Calculations Load or strain history Cyclic plasticity model Stress and strain tensor Search for critical plane Multiaxial Fatigue 2003-2013 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 Automotive Applications Presented at SAE 2004. Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 102 of 134

Biaxial and Uniaxial Solution 10-2 10-3 10-4 Uniaxial solution Signed principal stress Critical plane solution Damage 10-5 10-6 10-7 10-8 1 10 100 1000 10000 Element rank based on critical plane damage Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 103 of 134

Nonproportional Loading Summary Nonproportional cyclic hardening increases stress levels Critical plane models are used to assess fatigue damage Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 104 of 134

Notches Stress and strain concentrations Nonproportional loading and stressing Fatigue notch factors Cracks at notches Multiaxial Fatigue 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 106 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 108 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 110 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 113 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 114 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 118 of 134

Maximum Tensile Stress Location σ σ τ = σ τ 32 45 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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 120 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 121 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 123 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 124 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 125 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 127 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 128 of 134

Multiaxial Fatigue 2003-2013 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

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 131 of 134

Multiaxial Fatigue 2003-2013 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

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 133 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 134 of 134

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 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 135 of 134

Multiaxial Fatigue